If they have a more conservative third party candidate to vote for they may.
From the Pew Research Center:
The GOP nomination race among Republican evangelicals, in particular, appears to be wide open: Giuliani, McCain and Thompson each draw about 20% of the vote among white Republican and Republican-leaning evangelical voters, with Huckabee and Romney getting about 10% each.Good news for The Constitution Party bad news for Giuliani and the GOP. However I have to wonder about the "Hillary effect" coming into play. By that I mean will having a common enemy cause GOP supporters to rally around the Republican candidate just to keep Clinton out of power? I think Giuliani is counting on it.
In addition, a solid majority of Republican white evangelicals (55%) say they would at least consider voting for a conservative third-party candidate if the general election is between Giuliani and Clinton. Overall, 44% of Republicans and Republican-leaning voters say they would consider backing a third-party candidate who holds more conservative positions than Giuliani on social issues like abortion and gay marriage.
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