The standoff between Russia and Turkey intensified on Friday, as the Kremlin snubbed Turkish President Recep Tayyip Erdogan’s calls for dialogue and reiterated demands that Turkey apologize after its forces downed a Russian Su-24 bomber on the Syrian border.

Russia’s unforgiving response to the Nov. 24 incident—the first time a North Atlantic Treaty Organization member has shot down a Russian warplane since 1952—has plunged relations between Moscow and Ankara into uncharted territory.

It...