Republican like the Constitution except when it conflicts with their ideology. As a case in point, the refusal of Republican governors to accept Syrian refugees. Regardless whether or not bringing potential extremists into this country is prudent, the Constitution is rather explicit where this power lies. The power lies with the federal government.
This means that these governors who proclaim their love for the Constitution are actually thumbing their nose at it. They do not have the authority to do so. To prevent refugees from coming into their state is not based on legal precedent.
While we should be making sure that we are not importing extremists into this country, this is not an issue that can be resolved at the state level. It needs to be done at the federal level. If the Republicans feel that Obama is not operating in the best interest of this country, that pressure needs to come from their majority in Congress.