no the radiator cap does not regulate the heat. It regulates the pressure which builds with heat. when pressure rises to what the cap is rated for the spring compresses and alows coolant to flow into resivoir. Any how with out the cap most likly u lost most of ur coolant which causes engine to run hot and blow the head gasket and or warp and crack the cylinder head.
The problem is that you do not use the radiator cap, which regulates the heat. The loss of fluid eventually causes overheating and it can result to bad damages. I would advise you to get a radiator cap and keep changing it regularly. Contact a mechanic, for more consultations.