Relationship between the size of CPU registers and main memory

I have read this post however I still have a question.

I understand that saying 32 bit processor implies registers of size 32 bit long each of which can hold up to 2^32 data. My question goes little bit further. So during the execution cycle, how does 32bit register span into the main memory? In order word, does it mean that each register can deal with 2^32 memory addresses so that any instruction in this region can be fetched in one clock cycle?