TEL AVIV—The Trump administration said it would merge its Jerusalem consulate responsible for relations with the Palestinians into its newly relocated U.S. Embassy there, another symbolic blow to American-Palestinian relations.

The consulate in Jerusalem has functioned essentially as an embassy to the Palestinians. It was separate from the operations of the U.S. Embassy, which stewarded relations with the Israelis from Tel Aviv until May, when President Trump moved it to Jerusalem to fulfill a campaign promise.

Secretary...