Proly not.
If you're absolutely determined to get web content from within an SQL environ, there are as far as I know two possibilities:
Write a custom MySQL UDF in C (as bobince mentioned). The could potentially be a huge job, depending on your experience of C, how much security you want, how complete you want the UDF to be: eg. Just GET requests? How about POST? HEAD? etc.
Use a different database which can do this. If you're happy with SQL you could probably do this with PostgreSQL and one of the snap-in languages such as Python or PHP.
If you're not too fussed about sticking with SQL you could use something like eXist. You can do this type of thing relatively easily with XQuery, and would benefit from being able to easily modify the results to fit your schema (rather than just lumping it into a blob field) or store the page "as is" as an xhtml doc in the DB.
Then you can run queries very quickly across all documents to, for instance, get all the links or quotes or whatever. You could even apply XSL to such a result with very little extra work. Great if you're storing the pages for reference and want to adapt the results into a personal "intranet"-style app.
Also since eXist is document-centric it has lots of great methods for fuzzy-text searching, near-word searching, and has a great full-text index (much better than MySQL's). Perfect if you're after doing some data-mining on the content, eg: find me all documents where a word like "burger" within 50 words of "hotdog" where the word isn't in a UL list. Try doing that native in MySQL!
As an aside, and with no malice intended; I often wonder why eXist is over-looked when people build CMSs. Its a database that can store content in its native format (XML, or its subset (x)HTML), query it with ease in its native format, and can translate it from its native format with a powerful templating language which looks and acts like its native format. Sometimes SQL is just plain wrong for the job!
Sorry. Didn't mean to waffle! :-$