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Is it grammatically correct to use "such as" the following way?

Many universities, such as UC Davis, UC Santa Cruz, and UC Irvine, are now offering degrees in environmental engineering.

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Yes, it is perfectly fine. "Such as" is essentially a different way of saying "for example".

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This is perfectly grammatical, but I find this usage of such as to be a little stilted. I'd suggest instead

Many universities, including UC Davis, UC Santa Cruz, and UC Irvine...

One could also use among them, or to emphasize a little more strongly that these particular schools have programs (rather than simply giving them as co-equal examples) one could write,

Many universities---UC Davis, UC Santa Cruz, and UC Irvine included---...

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Your example is correct, although from reading the question, I initially thought you meant that you wanted to actually start a sentence with the phrase "Such as..." and I can't really think of a good way to do that.

Depending on the larger context, I'd consider changing your sentence to:

Degrees in Environmental Engineering are now offered by universities such as UC Davis, UC Santa Cruz, and UC Irvine.

Why? It seems like the focus of the sentence is on the degrees, not the universities (which seem to be almost relegated to a parenthetical aside.) I like putting the important part first. I tend to be direct in my writing, which tends to be for legal/technical audiences. Your mileage may vary.