"Yes, very much," is a grammatically correct response, but can one say, "Yes, really much," and if not, then what are the differences in usage between very and really?

There's a difference between gramatically correct and meaningful, and between gramatically correct and common usage. 'Very' indicates intensity, while 'really' strictly indicates that contrary to expectations something IS of a certain description. As with terrible, awful and nice, common usage has eroded this distinction. However, 'really much' is not correct common usage. 'Much' is normally only qualified by 'very' or 'not' whether standing alone or in a phrase ('Very much ado about nothing' would be quite correct). In English, the absence of a rule to say you can't doesn't mean you can.

What it comes down to as well is that "A real lot" or the use of real to modify the quantity of something as in "I like you a real lot" is COMPLETELY incorrect. "really much" is a phrase that is used incorrectly to indicate the level or amount of "much" in this case..