BEIRUT—A Kurdish-led military force backed by Washington and now approved by Damascus is closing in on Islamic State’s stronghold of Raqqa after taking the strategic Tabqa dam nearby.

The Kurd-dominated Syrian Democratic Forces captured Tabqa Wednesday, a day after the U.S. pledged to arm the fighters. On Monday, the Damascus government for the first time endorsed the group’s battle against Islamic State, with Syrian Foreign Minister Walid al-Moallem complimenting the SDF’s fight against Islamic State at a press conference...