A bomber on an active combat mission is aiming for the Turkish national border. Turkish flight controllers issue warnings to that aircraft that their course needs to change. No response is received and the aircraft is not changing course. Turkish fighters are scrambled to intercept the combat aircraft heading for their national border. Additional warnings are issued by Turkish flight controllers.
Turkish pilots are informed by flight controllers that the bomber is still aiming for Turkey's border. They issue warnings on guard that the flight needs to change course immediately. One would assume that every system on the Russian aircraft designed to detect a targeting lock is going off, though the weapons officer on the Russian bomber reports that they had no warning that they were being fired upon - really? Russians can't detect an F-16 weapon system targeting them?
Additional warnings are sent, then the Russian warplane enters Turkey and the Turkish pilots fire upon the bomber.
So, to address your question: Do I think a nation (any nation, I honestly don't care which, could be North Korea for all I care) has the right to shoot down a combat aircraft that has entered their airspace and refused to reply nor change their course? Absolutely.
The claim is that they were going to bomb a target in Syria - so they notified Turkey of their intended target? Provide that evidence, Russia. Provide anything that would fully justify purposefully violating Turkish airspace with a combat aircraft in active combat operations.
The claim is that the Russian pilot never heard the warning - recordings from civilian flights in the region recorded the entire guard warnings from start to finish. Even if the aircraft in question somehow didn't hear the warnings, Russian controllers absolutely did, and unless they have no communications capabilities, they had every option to warn the pilot to change course.
This was yet another intentional action of violating NATO airspace on a combat mission. Russia's never shown any fear of doing that.
Oh, but Turkey's violated Syria's airspace repeatedly while bombing Kurds - and? If Syria shot down the Turkish bomber in their airspace, I've no problem with that.
We're not talking some civilian flight straying off course, or even a military supply craft - it was a warplane on active combat operations, and last I checked, warplanes don't broadcast their targets for the world to see.
I'm very happy at least one person lived through this case of hubris. Hopefully the Russians won't kill any more of their people by not responding to standard communications.