raise the public awareness of/about

Yes, you can. I would also drop the definite article, i.e.: "We should raise public awareness about environmental protection."

I have no opinion about which is "better," but i would probably naturally use "about," myself.

No. "Of" is the right word.

Contrex is wrong. Using "about" is acceptable, even if he personally does not agree.

So far I have found two persons said "of" while two said "about", then I am entirely confused!!!

I would say that "aware of" has a much stronger collocation than "aware about" but would consider neither incorrect, and at most the uncommon collocation might be awkward.

USA region:

Results 1 - 10 of about 65,400,000 English pages for "aware of".

Results 1 - 10 of about 287,000 English pages for "aware about".

Results 1 - 10 of about 33,700 English pages for "aware regarding"

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UK region:

Results 1 - 10 of about 3,010 English pages for "aware regarding".

Results 1 - 10 of about 31,900 English pages for "aware about".

Results 1 - 10 of about 5,070,000 English pages for "aware of".