The President may, with the consent of two-thirds of the Senate, make treaties with foreign nations. Otherwise the President doesn’t have the power to unilaterally make a treaty with a foreign government.
Here is the pertinent clause (Art II, Section 2, clause 2:
2: He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur;
note the word “present”. That’s how they slide crap through.
http://constitutionus.com/#a2s2c2
This is an indexed Constitution. I love it.
Obama can’t make a treaty on his own, but what Obama and any other President can do is cause US Policy to be more favorable or less favorable to any particular country without engaging in any treaty. Of course “policy” only lasts as long as that particular President is in office.