There are Muslims who wish to accomplish the goals of Islamic Jihad peacefully. Would this be an acceptable alternative to violent Jihad? We cannot answer this without a thorough understanding of the goals of Jihad.
What Are The Goals of Jihad?
The global Jihad movement takes on a number of different forms in different local environments, but there is one overriding goal of Jihad: to remove all un-Islamic regimes from power and install Islamic government, ultimately to be united under a global Islamic Caliphate. Technically, it is against Islamic law for offensive war to occur against non-Muslims without the authority of the Caliphate; however, the Jihadists define “defensive war” loosely enough that it’s kind of a moot point. Any grievance can justify defense, and who doesn’t have a grievance? The Jihad movement takes advantage of local grievances; they “think globally and act locally”. There is a great deal of evidence that this is true. 
What is an Islamic Caliphate?
The Caliphate would be the Islamic world if ruled by a Caliph. There was an Islamic Caliphate from the early days of Islam until early 20th Century, although from time to time there were some areas of the Islamic world that were ruled independently of the Caliphate. For example, in recent times, parts of Asia were on their own. The Caliphate was abolished in 1924 by Ataturk, a democratically-minded ruler of Turkey. In 1928, the Muslim Brotherhood was founded in Egypt, with the goal of re-establishing the Caliphate. Once the Caliphate is established, the goal would be to subjugate the rest of the world under Islamic Law. 
What is Islamic Law?
The implementation of Islamic Law (Sharia) has varied from time to time and place to place. However, even when the implementation was weaker, the more severe versions of Islamic Law were never removed from the books, and haven’t been to this day. There are four major schools of Sunni Islam, which encompasses the majority of Muslims. There is a consensus of the four schools on 75% of their legal conclusions. These schools were founded about 1000 years ago, and have remained basically unchanged to this day. The reason for this is that about 900 years ago, “ijtihad”, which means “free thought on religious matters”, was closed down. It was then declared that everything of importance had already been decided, so any further innovation would be heretical. This decision also marked the beginning of the end of the Islamic Golden Age, and since that time there has been very little development in the Islamic world in areas of philosophy, mathematics, medicine, science, or literature.  Today there are brave reformers, such as Irshad Manji, who are trying to open the gates of ijtihad, but their influence would be difficult to discern at present.
Here are some common elements of Islamic Law:
Apostasy: Anyone born to one or both Muslim parents is required to be Muslim. Any Muslim, whether by birth or conversion, is required to remain a Muslim. For a Muslim to leave Islam is punishable by death, if it’s a man. This is agreed by all four schools of Islamic Law. One of the four schools would give the same penalty to a woman; the other three would give her life in prison. 
Freedom of expression: It is illegal to criticize Islam, Mohammed, or Sharia. Attempts at reforming Islam or Sharia can be called “heresy”, which is illegal. Attempts at converting Muslims to other religions is illegal (but Muslims are free to proselytize). 
Women: Women are treated as minors their whole lives. They are always under the protection of their fathers, brothers, husbands and/or sons. They may be married off at a very young age, even before puberty, with no say. Their testimony counts as half that of a man’s in court. Their inheritance is half that of a man’s. They can only prove rape with four pious male witnesses; otherwise, to allege rape could get them punished for illicit sex, which is the woman’s fault. There’s no such thing as rape in the context of marriage. Wife-beating is clearly sanctioned in the Koran. A man can divorce his wife by saying “I divorce you”, in which case he has custody rights; a woman cannot divorce her husband. Although it is legal, at least in some circumstances, for a Muslim man to marry a non-Muslim woman, the reverse is not legal. Because Muslim men are allowed to have up to four wives, this potentially puts more of the child-bearing capacity under Muslim control. It is also legal for a Muslim man to have sex with his female slaves, whose marriage (if any) is instantly dissolved upon capture. (Slavery has been abolished throughout much of the Muslim world, but continues to be practiced in some countries.) Abortion is most likely illegal. Then there’s the headscarf, or other means of coverage for women. 
Non-Muslims: A non-Muslim’s testimony in court is inferior to that of a Muslim. Non-Muslims cannot practice their faith openly, or display religious symbols. Non-Muslims cannot hold a position of power over Muslims, as in government. Non-Muslims have to pay extra taxes (this one is required by mainstream interpretations of the Koran, see 9:29). There have been poll taxes, property taxes (since they’re really just renting the land from the Muslims, rather than owning it), and sometimes double the regular taxes that Muslims pay. Treatment of non-Muslims has varied from time to time and place to place, but Islamic Law is very clear on the principle that they are second class citizens. Their payment of extra taxes is in exchange for their lives being tolerated (protection money). At times, a group of non-Muslims has been singled out for especially harsh treatment. Here are a few extreme examples: As recently as the 20th Century, Yemen has had a law that Jewish orphans must be forcibly converted to Islam. Iran once had a law that Jews could not go outside in the rain, since rainwater could splash from the Jew onto the Muslim, thus defiling the Muslim. There are many, many other examples of discriminatory laws toward non-Muslims that I am not listing here. 
Gays: Under Sharia, homosexuals would be killed.
Note that currently, very few countries live under the full Sharia; some experts say just Saudi Arabia and some also include Iran. This is one reason that Jihadists target Muslim countries: they are trying to make Muslim countries more Islamic. Worldwide, more Muslims than non-Muslims are victims of Jihad. Strategically, it makes sense for Jihadists to go after Muslims first, because after reaching their goals in Muslim countries, they would then have more potential recruits with which to go after non-Muslims.
It would also be useful here to mention that there are many examples throughout history of other religions persecuting their religious minorities and mistreating women. However, so far as I know, there is no other religion at present which has a global movement for implementing a legal system like Sharia, so fundamentalist Islam is really the only one which poses a current threat.
However, when I say “fundamentalist Islam”, don’t make the mistake of thinking this means a small fringe group. There is reason to believe that fundamentalism is really the mainstream of Islam; in any case, it’s not a tiny band of extremists, and many observers say it’s growing.
To What Degree Do Muslims Support Sharia?
This is a big unknown. This is partly because in the West, due to the impoliteness of saying anything critical of non-Christian religions, few people are even fully aware of what Sharia is. I think most Westerners think that it would be a good thing for Muslims to pursue Islamist goals non-violently, because they don’t really understand what Islamist goals are. This is one reason more education about Jihad ideology is needed: we need to nip the whole Sharia movement in the bud.
It is clear that not all Muslims support Sharia; perhaps even a majority in the US do not. In Canada, there was a referendum to impose Sharia family law on only the Muslim citizens, which failed. If this had succeeded, a baby born in one house would have had a different set of rights than a baby born next door. This referendum was defeated mainly by women. However, keep in mind the Islamists see themselves as part of a 1400 year old ongoing struggle that will only end when their goals are met. They do not see a tactical defeat as an end, merely a temporary setback.
There are some disturbing indicators of fairly widespread support of Sharia in the US. First, to my knowledge, no mainstream Muslim association or organization in this country has denounced Sharia. It is the small reform groups who have denounced Sharia.
Second, a fair number of Muslim spokespeople have said openly they do want to have Sharia replace the Constitution in this country one day.
Third, the first Muslim congressman in the US, Keith Ellison, who took his oaths of office on the Koran rather than the Bible, has refused to denounce Sharia.
Fourth, according to polls, a very significant number of Muslims have sympathy for the goals of Jihad.  So, even if it’s true that Islamic Jihad itself consists of a tiny fringe group, the same cannot be said for those who support the Jihad’s goals. If anything, I would expect the numbers indicated by polls to be understated, because it’s known that some people tell pollsters what they think the pollsters want to hear.
What to do about it?
This is a problem that cries out for more public discussion. I think it’s quite possible that some Muslims support the idea of Sharia because they feel it’s their duty to support it, perhaps without even fully knowing what’s involved. If so, then this, too, needs to be discussed.
In addition, the stigma needs to be removed from those who oppose Sharia, who are currently often branded “racist bigot hate-mongers”. Opposing Sharia is no different than opposing a Christian theocracy, for which there is currently zero threat, but lots of people opposing it. It is no different than opposing Communism, Fascism, or any other political system. Quite a few Muslim reformers want more non-Muslims to speak out against Sharia. It makes no sense to associate opposition to Sharia with hatred towards Muslims; on the contrary, Muslims would be among the biggest beneficiaries of the demise of Sharia, and many Muslims are already aware of this.
Some mainstream Muslim spokespeople will accuse those who oppose Sharia of being on a Crusade against Islam, but Westerners have no excuse for going along with such accusations. What it really boils down to is this: If Islam is just a personal religion and Sharia is not integral to Islam, then going against Sharia is not going against Islam. If Sharia is integral to Islam, then everyone who cares about individual rights should oppose it, and support those wanting to leave it or reform it.
On the other extreme, some Muslims insist, incredibly, that Sharia is not a system of law at all. These Muslims have a lot of explaining to do. If they are right, it should be no problem for them to convince all mainstream Muslim groups to denounce the implementation of the Sharia as a system of law.
 Front Page Mag: The Muslim Brotherhood “Project”, Patrick Poole
Daniel Pipes: What Do the Terrorists Want? [A Caliphate]
Front Page Mag: The Caliphate is Coming, Rachel Ehrenfeld
 Wikipedia: Caliphate
 Islam Watch: The Nostalgia of Islamic Golden Age vs. the History of Science, Syed Kamran Mirza
 Light of Life: The Penalties for Apostasy in Islam
Challenging Islam: What is the Shariah?, Mentat
 Ahmad ibn Naqib al-Misri, Reliance of the Traveller, Rev. ed. (Beltsville, Maryland: Amana Publications, 1994), 595-598, 609.
 Opinion Journal: Unfree Under Islam, Ayaan Hirsi Ali
Challenging Islam: What is the Shariah?, Mentat
 The Jizyah Tax:
Equality And Dignity Under Islamic Law?, Walter Short
Dhimmitude: History: Dhimmitude, Bat Ye’or
Challenging Islam: What is the Shariah?, Mentat
 Telegraph: Poll reveals 40pc of Muslims want sharia law in UK, Patrick Hennessy and Melissa Kite