I have zero knowledge when it comes to ships let alone military ones. Still, I don't understand exactly how fire can lead to sinking. How is fire capable of damaging the thick metal hull of a ship?
First of all, the fuel is flammable. I am not sure what kind of fuel Moskva used to use to power the gas turbines, the heavy diesel oil used on marine platforms burns just so good. Oil tanks are usually positioned underwater part of the ship but there is settling tank which is around the engine room so is the separators, lubricants etc. If any of those catches fire and if the ship is not equipped with proper fire-fighting equipment and suppressant systems in the engine room or wherever the fuel and lubricant oil fed from, then it continues to burn once set ablaze.
Secondly, the isolation material inside the hull, furniture, equipment. The fire propagates just nice through these parts. The fireproof doors, isolated compartments and sprinkles can stop this. At this stage it is related to fire-fighting equipment, fire-fighting plan of the ship and readiness of the crew.
Third is the ammunition but that rather becomes an explosion than a fire.
If any of these triggers then the steel - the material that ship is built - just burns, long before than it melts. Eventually reaches to ammunition storage and central magazines. In case of commercial ships the fire propagates and eventually weakens the hull enough to loose its integrity or reach to the underwater plating and the ship sinks by being flooded.
When steel burns the fire just propagates anywhere, freely.
