Not swearing oaths would be more christian, as would refraining from showy public prayers. Hypocrisy was a very big deal to Jesus. The bible makes it clear that He much prefers those who reject him entirely over those who claim to follow him but show otherwise by their actions. Obama does not rate well based on this, but neither do the vast majority of politicians.
I don't think it is very christian to oppose immigration, even illegal immigration. Abusing poor immigrants rather than showing them hospitality is the sin of Sodom.
Jesus is one who often took advantage of the Gleaner's Rights, which The Law of Moses explicitly guaranteed to aliens as well as the Israelites. It thus seems pretty unchristian to try to deny services to aliens that are granted to natives.
Supporting abortion (except when the life of the mother is in danger) is certainly unchristian. Christ may not be quoted as condemning it specifically in the gospels, but abortion is very explicitly condemned in the Didache, which may well be older than the gospels themselves. The church has always been pro-life. (It seems probable that Jesus would have endorsed the principle of Pikuach nefesh and so defended abortion in circumstances where it is deemed necessary to save the mother's life though.)
Perhaps the most unchristian thing Obama does though is wage war.
Funny... as an atheist, this is exactly how I feel. Swearing on a bible means NOTHING to me. So many people do it and they turn out to be utter liars. But my yes does mean yes, and my no does mean no. It's called word of HONOR. And I take my honor very seriously.
I remember reading about study a while back that found that swearing an oath on a bible actually does result in even atheists being more truthful. The same is true of being reminded of the 10 commandments, or of pretty much any sort of honor code though. Just thinking about a system of ethics, even one mostly rejects, leads to more ethical behavior.