Can the President effect that "on their own" or would it need some act of Congress?
By current precedent from the US Supreme Court, the US president not only has authority to recognize foreign nations, but exclusive authority. The pertinent case is Zivotofsky v. Kerry, where a couple wanted to list the birthplace of their child as "Jerusalem, Israel." At that time the State Department of the United States did not recognize Jerusalem as the capital of Israel, or in fact technically as part of Israel. Congress had passed a law allowing for precisely this case:
RECORD OF PLACE OF BIRTH AS ISRAEL FOR PASSPORT PURPOSES.—For purposes of the registration of birth, certification of nationality, or issuance of a passport of a United States citizen born in the city of Jerusalem, the Secretary shall, upon the request of the citizen or the citizen’s legal guardian, record the place of birth as Israel.
When the case went to the Supreme Court, this portion of the law was found to be an unconstitutional restriction on the power of the President to recognize foreign nations, and the power of the president to do so was reaffirmed.
Held: 1. The President has the exclusive power to grant formal recognition to a foreign sovereign. Pp. 6–26.
As such, it is within the US president's power to recognize the Golan Heights as being part of Israel, or not, without congressional support.