Don't use a regex. Use HTML::TokeParser::Simple and URI::Find (or URI::Find::Schemeless) instead. The HTML::TokeParser::Simple documentation has examples of very similar cases you can adapt. The core loop would look something like this:

my $finder = URI::Find::Schemeless->new(sub {
    my ($uri, $text) = @_;
    return qq{<a href="@{[ $uri->abs() ]}">$text</a>};
});

my HTML::TokeParser::Simple->new(\*STDIN);
while (my $token = $p->get_token) {
    my $text = $token->as_is;
    $finder->(\$text) if $token->is_text; # here's the key
    print $text;
}
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The key line invokes URI::Find::Schemeless only for tokens which are plaintext, not part of a tag or other things.

Makeshifts last the longest.