Your understanding is incorrect. This was an amendment to the NDAA. Basically the bill that Congress has to pass every year to authorize the government to maintain a standing army, and fund said army. Without the NDAA, our military would effectively cease to exist.
It would be impossible for President Obama to veto it.
True that he doesn't care per se, but as I doubt he's going to quietly retire to a Hawaiian villa in 3.5 years, he does have to gauge who thinks what of this presumably unexpected issue & backlash.
Also remember, there are still leaks to come, we've only seen a glimpse of the full scope of the revelations.