The US will not be pulling out of Qatar and there will be no escalation for the simple reason that there is too much at stake, should either or both of these happen.
The US stands to loose the most out of any country situated outside the Middle East, should things deteriorate further. Not only would they have a lot less influence in the region but there is currently around $20 Billion in defence sales with Boeing, that is pending agreement at present.
I suspect we will see a rapprochement in US vs Qatar relations and the US will walk the diplomatic tightrope by attempting to be cordial with all the GCC countries. Easier said than done I suspect!