Moscow said Russian and Syrian special forces had rescued one of the pilots who ejected from the plane after being shot down early yesteray but confirmed another was dead, in an incident that threatens to damage relations between two rival players in the Syrian conflict.
Russian President Vladimir Putin reacted furiously to what he described as a "stab in the back committed by accomplices of terrorists", recommending that Russians do not visit Turkey, a key tourist destination.
The shooting also risks derailing efforts to bring peace to Syria that were gaining tentative momentum following the November 13 Paris attacks claimed by Islamic State militants who control swathes of northern Syria.
Erdogan vowed to always defend Turkish borders but appeared to want to avoid provoking further one of the biggest crises between Russia and a NATO member in recent years.
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"We have no intention to escalate this incident. We are just defending our security and the rights of our brothers," Erdogan said in a televised speech in Istanbul.
"Hopefully, this is a moment in which all parties can step back and make a determination as to how their interests are best served," Obama said.
Erdogan and Obama agreed on the need to reduce tensions and prevent a repeat of similar incidents in a phone call late yesterday, the Turkish presidency said.
Following an extraordinary meeting of the alliance called by Ankara, NATO chief Jens Stoltenberg said "diplomacy and de-escalation are important to resolve this situation," he said.
He said both planes had flown 2.19 kilometres into Turkish airspace for 17 seconds from 0724 GMT yesterday.
Ankara and Moscow are already on starkly opposing sides in the over four-year Syrian civil war, with Turkey wanting to see the ousting of President Bashar al-Assad but Russia one of his last remaining allies.