My mechanic says the car overheated because the radiator cap was not screwed on tightly, and now he wants to replace the whole radiator. Is this really necessary?
Under most circumstances the system would vent the excess pressure via the pressure cap. If the cap is not installed correctly the system cannot vent. I could see where the radiator might be damaged if it was weak to start. Usually a hose or clamp will fail first. It is possible that your mechanic feels the radiator has issues not directly related to the incident that warrants it being replaced.
It could cause your radiator to leak, I have seen it happen. My past experience when people have over heated their vehicle it seems to make the radiator oxidize really fast. I had a family member who used to over heat their vehicle a lot and with in six months the radiator would be oxidized and leaking really bad. If you have a bad thermostat get it changed as soon as possible because it can create a lot of other problems.
Not only can the radiator get damaged, but the "Head" on the engine can warp and allow antifreeze to get into the oil which will ruin the engine. Check the radiator reservoir frequently to monitor the coolant level.