They do if the RNC is involved which it must be. Otherwise it would have to risk the possibility of certifying that a candidate is eligible for office of potus when they are not.
The democrats solved the problem by having Nancy P delete the paragraph that said O was qualified.
In the past the congress has verified when requested that mc Cain and others were qualified. This time McConnell said NO
Since in a Texas interview TC has said a natural born citizen has two parents who are citizens and they are born on US soil he can not claim he is not aware which in a suit against him would make a conviction of Fraud a certainty.
Furthermore his college professor who he praised publicly before this issue arose said he was in the original intent camp in college
It is possible that TC has laid his political life on the alter for the love of the constitution. His situation mimics Os particularly if a Kenyan BC has been found In that case he would certainly be a hero and we are all watching a well directed and written piece of drama
When you consider all of the legal avenues here, what you find is that the "natural born citizen" requirement laid out in Article II of the Constitution is unenforceable except by: (1) the voters, (2) the Electors in the Electoral College, or (3) Congress.
There is no role for a Federal court in the matter, which is why they have consistently refrained from making any decisions in these matters.
P.S. -- For an interesting bit of research on this subject, check out the disputed 1824 election and the events surrounding it. Interestingly, one of the candidates in that election -- Henry Clay of Kentucky -- did not meet the strict constitutional definition of a "natural born citizen" posted repeatedly here on FreeRepublic, but ran anyway. And from what I can tell, it wasn't even an issue among his opponents.