ISTANBUL — President Recep Tayyip Erdogan of Turkey lost his first major political battle with the Trump administration, which is arming the Syrian Kurds who the Turks consider enemies. The question now is what Mr. Erdogan, a headstrong leader, will do next.

The White House made the move to arm the Kurdish fighters, despite vociferous objections from Turkey, because it considers them an effective military proxy in the fight against the Islamic State.

But doing so comes at a cost. Angering Turkey risks a rupture with an important NATO ally that is being courted by Russia, and could have an unpredictable impact on the battle against the Islamic State and the wars in Syria and Iraq.

Mr. Erdogan and his aides have warned for months about taking more aggressive, though unspecified, actions against Kurdish militants — though in a different stronghold, Iraq. And analysts say such a plan would make some strategic sense.