Obama was clearly referring to multiple Republicans; if his remarks responded to any particular candidate, it was Chris Christie, who had earlier explicitly ruled out accepting "orphans under the age of five" for resettlement in America. Today, however, it was Ted Cruz who acted as if Obama had attacked him personally. "Mr. President, if you want to insult me, you can do it overseas, you can do it in Turkey, you can do it in foreign countries," Cruz said. "But I would encourage you, Mr. President, [to] come back and insult me to my face."
It seems clear from his statement that Cruz was ready to jump on any opportunity to start a fight with Obama, even if Obama had failed to do him the favor of mentioning him specifically. I have previously suggested that the popularity of Donald Trump and Ben Carson in the current Republican nomination race is a reflection of the formidable power of anti-Obamaism in the contemporary GOP. Cruz's potential path to the nomination almost certainly requires attracting a significant fraction of the Republican vote that is currently parked behind Trump and Carson, so it is in his particular interest to distinguish himself as an Obama antagonist in order to appeal to those voters if and when the Trump and Carson candidacies fade.
Of course, Cruz has not exactly been complimentary of Obama before today, but most of his energy since entering national politics has been devoted to fighting other Republicans, not Democrats. It will be interesting to see if today's remarks represent a larger shift in campaign strategy, with Cruz attempting to claim the mantle of the party's most anti-Obama candidate. If Obama wants to increase the probability that Cruz wins the Republican nomination, next time he'll attack the senator by name.