I don't think this situation could hurt President Obama but perhaps hurt. Now it looks like Romney's first response is to attack the president not issue concern and or have any empathy for the family of those that were killed. To remain calm and address the situation No he jumps the gun and looks foolish compared to all the other responses. Also this is right after a day that reminds us that Bin Laden was killed on Obama's watch and there hasn't been attack on the US while he's been in office. People aren't going to suddenly go "Romney would be better, etc". It looks like Romney is politicizing the tragedy.
This is similar to the 2008 Economy Crash where McCain suspended his campaign and Obama did not, the GOP is too reactionary and while it fires up their base most people will recognize that.
So far the response from many on the right should just go to show if Romney took office we would most certainly go to war or conflict with somebody in a matter of months.