Doubtful. It would be political suicide, as Democrats would dominate in elections nationwide.
You presume that libertarians still have a stake in the electoral success of the Republican Party. It is exceedingly clear to me that we do not.
A lot of libertarians and fiscally-oriented conservatives tend to vote Republican because in essence, their track record on economic issues has been much better than their track record on social issues. To clarify, taxes and welfare spending are still far lower than they were when Reagan took office. Not so much with abortion, homosexuality, secularism, etc. where by and large the status quo has prevailed OR we've moved in a more tolerant direction.
The one biggest, and most recent exception to that, is really the Bush years where domestic spending on things like entitlements
and massive roll backs in civil liberties both took place (both parties are to blame though). But even the Bush years were mostly a failure for the Christian Right's agenda, despite his strong support from them. People like Thomas Frank basically have it right, social issues are just a carrot the Republicans use to attract people that otherwise would be lukewarm or hostile to them.
Now could the GOP survive without us? I'd say they could, but it would require taking a more moderate to liberal stance on economic issues. Of course that would leave us fiscal conservatives and libertarians even more politically homeless than we are now.
