"Spell check" vs. "spelling check"
Solution 1:
I would say that I check the spelling of my papers by using the spell check function on my word processor. One describes the process, the other describes the computer function.
Solution 2:
Spell is recorded in the Oxford English Dictionary as a noun used colloquially to mean ‘a way or mode of spelling a word’. That would suggest that the answer to your question is not that spelling check is more ‘correct’ than spell check, but that it is more formal.