The Constitution is not self enforcing. In practice it is only enforced if the great majority of judges, politicians, and people working in the executive branch want it to be enforced. We have long passed the point where that is the case. When is the last time a politician (other than Ron Paul) refused to vote for a bill because it violated the 10th Amendment? When is the last time a state refused to use Federal Reserve Notes because that would violate Article I, section 10? Why should we trust the President to uphold the 2nd Amendment when in a conversation before he ran for president he said he didn't think Americans should be allowed to own guns? The progressives never minded those parts of the Constitution becoming dead letters. At least we have the 1st, 4th, and 5th Amendments, they'd say. Except now under the Patriot Act, the NDAA, the TSA, and the NSA, we don't.
Our public schools used to teach children about the Constitution (the written down one, not the "living" one). That last happened in the 1950s. How can you expect people to insist that their politicians uphold the Constitution when the vast majority of them have no idea what's in it?
So no, Obama is not "constrained by the system that our founders put in place." Neither is congress. And Obama knows this. That is the most dangerous lie that I ever heard a politician utter.