All three sentences are valid, though in different context and with different meaning.
A: "He looks at you [in] the same way Rick does." is how you would usually verbally say it, "it" and "as" are both superfluous in the sentence, and it rolls of the tongue more easily without them.
B: "He looks at you in the same way as Rick does." is the more grammatically correct sentence, but can become a bit stiff, due to all the extra words that doesn't carry any particular meaning.
C: "He looks at you as Rick does." has a different meaning than the other two. Here it's only stated Rick is looking at you, omitting how they are looking at you (lovingly, hatefully, reproachfully etc.) So here the two people can be looking at you in completely different ways, where sentence A and B the way they look at you are the same.
In some cases, it would be fine to substitute "the same way" with "as". An example would be "Do it the same way I do" that could also be "Do it as I do". However in your case the meaning of looking is actually changed from observing you, to instead put emphasis on the emotions the person is showing when they are looking at you.