Mac won't load some web contents
I used to get this mainly on banking sites. It might be a certificate issue, or more accurately, the older browsers that run on ML not being compatible with the certs or negotiate secure connections properly with the sites (perhaps the server actively denies access due to browser version).

As ping and DNS works, the problem must lie at a higher level.

Unless you've a reason to run ML still (I've only just updated to ElCap due to various bits of software) then I'd update.