Turkish-born gadfly Cenk Uygur thinks so (or at least hopes so), and is trying to run for the Democratic nomination.
In 2012, Guyanese-born lawyer Abdul Hassan made a presidential bid and made the same argument in multiple suits in state and federal courts, losing each case.
While the natural-born citizen clause is not the most honorable clause in the Constitution, the 14th Amendment only clarifies that anyone naturalized in the US is a citizen not just of the US but of the state in which they reside, leaving "natural born" citizenship as a separate classification.
The Privileges and Immunities Clause prohibits states from abridging the Privileges and Immunities of U.S. citizens. But by the text of the natural born citizen clause, becoming President is not a general right of all U.S. citizens.
And the Equal Protection Clause says no state can deny to any person within its jurisdiction equal protection of the law, referencing state law, not provisions of the US Constitution.