How does this affect internet searches?

Potentially a search engine could provide results without knowing the actual query, although that would make it difficult to identify new trends or determine which results are/aren't being clicked, so it probably wouldn't be competitive as a general purpose search engine.

It would also be completely impractical, since the search engine would necessarily have to perform some sort of scan of its entire database to respond to every query (otherwise it would, in fact, learn something about the query).

Unless the database is also encrypted and provides the same homomorphism against reverse index queries.

In which case you are no longer talking about FHE. With FHE the server receives a ciphertext query and produces a ciphertext output, without ever seeing any plaintexts (not even the result of a reverse index query).

If the database was encrypted, you would need a different encrypted database for every user - again impractical.