I would say it is unlikely so long as you 'turned the car off immediately'. There are measures you can perform before the engine is back to a running condition and may save the labour of installing a new fan only to remove it when more serious problems become evident with a running engine.
A compression test on each of the cylinders will determine whether the head gasket portion that maintains combustion and exhaust stroke compression has been compromised.¹
A cooling system pressure test will determine if the head gasket portion sealing the water passages has been compromised. This should probably be performed after the overheating in any event and the whole system examined for leaks.
One thing that many people ignore after suffering a catastrophic overheating event is removing the valve cover and re-torquing each of the head bolts in proper sequence. Overheating can change the torque on these very important bolts.
¹ A blown head gasket will typically leak rapidly; slower loss of compression may indicate valve or piston ring problems.