Well for the second part, Imma say no, just no. You can't declare that the constitution is unconstitutional.
As far as the ramifications, I don't see any serious problem with allowing naturalized citizens to become president, though most likely there should still be a certain time limit to keep somebody from becoming a citizen and running a week later. However the latter scenario I find unlikely, unless there's also a massive influx of immigrants at the same time intended entirely to support said candidate, and this is likely to be fairly obvious.
I would actually lay long odds that the current democratic party would act against allowing though, because if you allow naturalized citizens you're going to have Arnold Schwarzenegger running for president soon after. He basically only quit winning everything in politics because he'd hit all his term limits in California and couldn't reasonably run for any higher office after. And... none of their candidates are going to win against him.