You keep using that word... impeachment. You do remember when Bill Clinton was impeached, yes? Maybe not. But he was! And then continued on being president for two years, because that's not all there is to it.
Now, impeached and *convicted*, well, that's a different story, but that also has to go through the Senate. Clinton got a 50-50 vote on the obstruction charges, I believe, and that allowed him to remain president.
So if there is a conviction?
And therein lies the delicate balance of selecting your running mate. They can't be so abhorrent that they actually cause people to vote against you, but they also can't be anyone that people would more want as the president.
Of all the jobs the Vice President has, assassination insurance is the most unsung.