Mate/pal work just fine for ladies as long as they actually are your friend: best of mates, we're just mates, pals together and so on, but, mate/pal doesn't work as well as it does with men when the woman you are apologising to is unknown to you.
In my opinion, the safest Sorry (for a man) to say to a woman is just plain Sorry.
The following terms are UK based.
If you want to take a little risk then you could try
Sorry Miss for a youngish lady or Sorry Madam for an older one but that seems quite stuffy and Madam might not always be well received.
I don't think Sorry Missus is very popular unless said by children.
All the phrases below are also used by women when talking to women or men (friend or stranger).
Sorry love is generally acceptable in most of the UK.
Sorry dear should perhaps be reserved for ladies older than yourself these days, but then if they don't think they are old ...
Sorry darling might well get you some grief unless the lady is your wife, girlfriend or a friend although it was (and probably still is) commonly used in the South.
Sorry pet seems reasonably acceptable in Northern England along with Sorry lass if the lady is younger than you.
Sorry hen in Scotland seems to cause offence as often as it doesn't.
This probably applies equally to Excuse Me: when you want to interrupt someone to ask them a question Excuse me mate works fine when you are asking a man, just plain Excuse me when you are asking a woman.