But Obama can unilaterally - without the intervention of the Party of No! - suspend the operation of this provision under 10 USC 12305 - the stop loss provision. (See this PDF.)
So the opposition of the Party of No is clearly not what is holding him back.
Same for JFK and what eventually became his EO 11063, desegregating new Federally funded housing. He could have issued the EO on Day 1, but wanted to stroke the Southern gatekeepers of the Congress. (Not that that did him much good, of course.)
In the event, it came on November 20 1962.
Back to Obama: shows how useful the Congress is to him - another thing he and his (and their) corporate sponsors have in common.