Some operations can be shortened even further. It is up to the reader to decide if that increases or decreases maintainability :)

sub match_url {
    my $row = shift;         # No @_ required, that is the default
    my @matches = ($row =~ m/$re/g) or return;
    # do something with @matches
    }

if (!@ARGV or $ARGV[0] eq "-") {
    chomp (@ARGV = <STDIN>); # chomp can be done in one go
    match_url ($_) for @ARGV; # Your match_url takes one arg, not a li
+st
    }
else {
    # Do I need to handle the error case with this syntax?
    while (<>) {
        chomp $_; # Is chomp really needed? match_url probably ignores
+ trailing EOL in $re
        match_url ($_);
        }
    }
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for vs while

($str =~ m/$re/g) (note the parens) returns all matches in one list

use 5.012003;
use warnings;

my $str = "abacadabra";
my $rex = qr{(a\w)};

say "-- say";
say for ($str =~ m/$rex/g);

say "-- while";
while ($str =~ m/$rex/g) {
    say $1;
    }

->

-- say
ab
ac
ad
ab
-- while
ab
ac
ad
ab
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