No, by shouldn't be omitted after impressed.
In writing, people generally use by, with, or that after impressed
I'm impressed by his skill
"President Obama Is Not Impressed With Your Right To Modify His Photos"
I'm impressed that it's selling so well this soon.
A sentence like "I'm impressed how tall he is" doesn't sound like idiomatic native-speaker English to my ear, but I'm sure that people say and write it. A good copy editor will change it to "impressed by how tall" or, perhaps, "impressed with how tall", or "impressed that he's so tall".
Google Ngrams shows that
1 impressed by
2 impressed with
are most frequently found in books,
3 impressed that
is rarely found in books, and
4 impressed how
is never found in books. This doesn't mean that people don't say it, however: people will say anything, regardless of whether it's right or wrong.
Perhaps choosing between by and with is merely a matter of style. You'll have to decide based on context and the specific sentence whether one is better than the other. Give us some choices and we'll see what everyone else thinks -- maybe.