Is the US Pledge of Allegiance a form of oath or vow, and if so may one recite it? If one has recited it has one incurred any new halachic obligations?
If you say it but don't intend it, it doesn't count as a vow (Yoreh Deah 210:1). However, the Bach and Maharshal (quoted in Be'er Heitev 210:1) say that if he intended to misspeak, what he says counts. But if you were forced to say the Pledge of Allegiance (as in Yirmeyahu's boy scout example), you are allowed to intend in your heart for it not to count (232:14).
If you already said the pledge of allegiance, I don't know of a way that it couldn't be a valid swear — but there always is hataras nedarim, of course. To do that, though, you have to regret it, and if you had known something you wouldn't have sworn. So if you are going to spy for Russia, you can say "If I had known I would be spying for Russia against the USA, I wouldn't have sworn," and get three people to revoke the neder (but then again, if you were forced to work as a spy for the Russians, it counts as a "neder ones" (Rama, Yoreh Deah 232:12)).
Alternatively, if the USA stops giving "liberty and justice for all," the neder is not valid and doesn't even need to be revoked, because you vowed under the impression that "the republic for which [the flag of the United States of America] stands" was "one nation, under G-d, indivisible, with liberty and justice for all." This applies even if you didn't say "on condition" (228:19).
I am afraid I do not have a written source on this but I asked this sheilah (question) l'maaseh (for practice) regarding the Pledge of Allegiance and the Boys Scout pledge, since I have always been uncomfortable with such things. I was told since one may at any time quit the organization, it was permitted. The idea as I understand it is that insofar as it is ultimately a voluntary "obligation" that one may retract at will there is no problem. While this answer is most directly and practically applicable to the Boy's Scout pledge, the wording of the Pledge of Allegiance has the same leeway. While a good-faith recitation of the Pledge implies that it is more than a momentary affirmation it is not worded or intended as a vow permanently surrendering oneself to the state without qualification or restriction. A practical example is the Pledge doesn't preclude one from one day deciding to make aliyah.