Probably not... So long extremist Islamic groups target infidels exclusively, there will still be support for those groups amongst majority of Muslims.
Most Westerners assume that only a minority of Muslims are Islamic extremists, or that an extremist minority of Islamists use violence and intimidation to force the moderate majority of Muslims to accept the minority's dominance. But the American polling company Pew Research found in a 7-country poll last year that most Muslim Egyptians, Jordanians, Pakistanis, and Nigerians, and a large minority of Muslim Indonesians back 3 extremely violent Sharia law punishments: 1) stoning adulterers to death, 2) cutting the hands of thieves, and 3) execution of people, who abandon Islam. And in two relatively Westernised countries, Turkey and the Lebanon, where Islamism's influence is weaker – those punishments were also quite popular with 16% and 23%, respectively, supporting stoning of adulterers, 13% supporting hand-cutting, and 5% and 6% supporting execution of apostates.
Moreover, a poll of predominantly Turkish men by a Turkish University in 2005, as reported by the BBC, found: 37% supported killing of women, who commit adultery, and another 21% supported having their noses or ears cut off. So, one should wonder how moderate a country Turkey is?
Extreme and barbaric punishments, like stoning-death for adultery or chopping of hands and feet etc., are part of Sharia law. And this 2010 Palestinian poll showed that most young Palestinians want to live under Sharia. Like most Palestinian Muslims, most Muslim Egyptians, Jordanians, Pakistanis, and Nigerians, and possibly most Turks also favour extremely violent punishments, given a February 2005 University of Jordan report, "Revisiting the Arab Street" (p. 53), reported that most Palestinian as well as most Egyptian and Jordanian Muslims wanted Sharia to be their country's only legal system.
Given such huge popularity of Sharia amongst Muslims in these Islamic countries, it is not surprising that the "freedom fighters" of the Libyan National Transitional Council (NTC), after ousting Gaddafi from power, immediately announced that Sharia, including polygamy, would be introduced in Libya. So, during the Libyan conflict, perhaps journalists shouldn’t have called anti-Gaddafi militia as "freedom fighters", who would bring "a new dark age for the Libyan women". In the battle between the Gaddafi regime and the NTC, there were no "good guys".
That fact was proved by this "Independent" article, "Leaked UN report reveals torture, lynchings and abuse in post-Gaddafi Libya".
In an August 30, 2007 article, "The women of Iraq", for Radio Netherlands, Nicolien den Boer revealed that since the fall of the tyrannical but rather secular Saddam Husein, Islamists have begun to force Iraqi women to dress more modestly, and because of the Sharia, the new post-Saddam Iraqi Government legalised child-marriages involving men and 9 year old girls; so a new dark age for women has also begun in "liberated" Iraq.
Even in Tunisia, also called a "moderate" Muslim country like Turkey and Lebanon, which by Arab Muslim standards it is – Islamists got highest proportion of votes (38.90%) in the October 2011 election, and their leader declared on November 13 that he hoped the election victory would lead to a new "caliphate", i.e. a royalist dictatorship. Establishing a (global) caliphate is Al-Qaeda’s key aim. So, even in the 2nd most Westernised Arab nation after Lebanon, Muslim voters have embraced a key totalitarian Al-Qaeda goal.
Westerners, who see the Egyptian uprising as the dawn of a new Western style democratic era in that country, may thus be in for a huge shock, as mostly pro-Sharia Egyptian Muslims are more Islamist minded than Tunisian Muslims.
Pew Research have also found that although support for Al-Qaeda continues to go down in Muslim countries (they have been regularly asking Muslims in 6 countries, and in the Palestinian territories about their views on Al-Qaeda and Osama bin Laden since 2003, and in Egypt since 2006), in 2010, 48% of Nigerian Muslims still had a favourable view of Bin Laden (down from 61% in 2006), and 49% still supported Al-Qaeda.
World leaders like Tony Blair think that the decline in support for Al-Qaeda was a sign that Islamic extremism was declining, but in actuality, Al-Qaeda and other like-minded Islamic groups’ engagement in bombings and carnage in Pakistan, Egypt, Indonesia, Jordan and many other Islamic countries, which also killed large number of Muslims, is the reason for the decline in support for Al-Qaeda amongst Muslims. Had al-Qaeda, and other extremists Islamic groups motivated by al-Qaeda, exclusively targeted infidels and Westerners, Muslims would still be supporting those violent Islamic groups as before. Indeed, as this 2009 BBC News story explains, many Arab Muslims pray every week in mosques for the destruction of all non-Muslims, which amounts to calling for the death of the world 5.5 billion non-Muslims.
And now that the Nigerian Islamic terrorist group Boko Haram, backed by Al-Qaeda, have started bombings in that country, killing Muslims as well, you can expect to see a much bigger decline in Al-Qaeda's support among Muslims in that country, if Nigerian media report that Boko Haram is backed by Al-Qaeda. So, in actuality, support for extremist violence in Islamic countries is not lessening; instead it might well be getting worse, given a growing numbers of Muslim women taking to veiling, a Sharia-compliant mode of dressing, which Islamists want to impose.