I don’t think so.
Here’s what the Constitution says-
“No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President;”
It doesn’t say anything about eligibility with regard to running for the nomination of a party, nor does it say anything about eligibility while running for president.
It only seems to reference eligibility to the Office of President, which to me means the foul occurs once he wins the election, but up until such time as he wins, he has done nothing that couldn’t warrant a suit.
In short, there doesn’t seem to be anything that says Americans can’t nominate someone who is ineligible or even elect someone who is ineligible, but that an ineligible person cannot assume the Office of President.
Which makes me think that he could go all the way, win, and then be disqualified by a suit, and that is not something that I find appealing to be perfectly honest.
If that were to happen, who would then become President ?
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I do think that Cruz could go all the way, I doubt that he will, I think Trump may have it sewed up. If Cruz wins the speaker of the house certifies the election and then he is president. True there might be lawsuits but they won't affect his being president until it is appealed to the Supreme Court which could take years. I have my doubts about a court having the guts to kick out a sitting president.
I am a Cruz first supporter. I wish however that he had renounced his Canadian Citizenship a long time ago instead of waiting until he decided he wanted to be president.
I do not believe that any of the lawsuits will prevail against Cruze but it will certainly make a circus attitude with the press. The press was afraid to say anything on the Obama birth issue but won't feel any constraint with Cruze. If nothing else it will be interesting and entertainging.
In fact, "being nominated" and "running for President" have no constitutional existence. Both of those quaint customs could be done away with tomorrow, and the Constitution would remain undisturbed.