Actually, he lost the popular vote by over 100,000, so that would have been a tough road to hoe, pragmatically speaking. Otherwise, Im sure Nixon would have done his darndest.
Not to prolong this, but Nixon actually won the popular vote totals, also.
It's a little known fact that some of the votes in the South (specifically Alabana )were not specifically for Kennedy but for an unidentified Democrat. The news media routinely put them in Kennedy's column but the story is more complicated than that. See the Wall Street Journal article, linked below. Nixon actually won the popular vote.
http://www.opinionjournal.com/diary/?id=110004320