Is the statement: "If Allah brings me this, I will stay away from masturbation" considered a vow, and is expiation obligatory if it is violated, knowing that the speaker did not intend it as a vow at the time, forgot what he said, and committed masturbation once?

As long as the questioner did not intend or vow a nadhr (vow), what he said is not considered a nadhr, because the mentioned phrasing is not among the explicit forms of a nadhr. He is only obligated to repent from that habit. And if we assume that he did intend a nadhr, then his nadhr is a nadhr to abandon an obligation, which is not binding according to the majority of scholars. Therefore, nothing is entailed by not fulfilling it. However, Sheikh al-Islam Ibn Taymiyyah held the view that an expiation for an oath (kaffarat yameen) is obligatory if it is not fulfilled.