Turkey and Russia can get over this but it will take serious, level headed diplomacy.
This appears to have already begun. The scheduled meeting between the two states (and Iran) has not been cancelled. This gives the Turkish side an opportunity to make fulsome apologies, fierce condemnations, and blame the Gulenists for the murder. The Russian side can gracefully accept the apology and move on attempting to take higher moral ground than before.
The man identified as Andrey Karlov’s murderer, 22 year old Mevlut Mert Altintas, was a police officer and had according to the authorities, already been investigated for alleged involvement in the failed coup in Turkey this summer and connections to the Gulen movement. Ankara can apologise for incredibly lax security, and if he was indeed still a serving officer be shamefaced about having a terrorist assassin they knew nothing about on the inside of the force. Ankara is now forced into telling Moscow it is trying to deal with its own ‘revolutionary’ movement while using the incident to continue the crackdown on the Gulenists and other opponents.
Russian Ministry of Foreign Affairs spokeswoman Maria Zakharova has already said “We are currently in contact with Turkish authorities who are assuring us that a thorough, comprehensive investigation will be conducted.” There has not been a hint of Russia suggesting this was an act by the Turkish state. If one comes, it is theoretically a step towards military action, as the state assassination of another state’ ambassador can be considered an act of war. Turkey’s President Erdogan has already headed off even the unlikely possibility of Russia accusing Turkey of ordering the killing by saying “This provocation is aimed at damaging normalization of ties with Russia’.
The shooting does look like an attempt to drive a wedge between the two countries following the slow rapprochement which followed the initial angry diplomatic exchanges in the aftermath of the shooting down of a Russian jet on the Syrian/Turkey border in November 2015. The two sides subsequently re-instated suspended economic ties, and now, with their military forces operating so close to eachother in Syria wise heads know they need to co-operate.
If they can get over the downing of a warplane, they can get over this. Neither side wants war with the other. That does not mean a war is impossible, but it gives diplomacy a head start, and a good chance of success.