As a native french speaker, your version wouldn’t sound really odd to me, nor be totally inappropriate.

But just as a note, it may be a slightly sad or depressive way for the text to end. A variant would be :

J’aimerais beaucoup faire le gig, mais, malheureusement, je ne suis pas disponible le week-end du 20 juillet.

Désolé et bonne continuation, (if there’s few chances you’re talking to them again)
(or)
Désolé et à bientôt, (if you expect to see them again of course)
(or even)
Désolé, la prochaine fois j’espère, (another friendly-sounding alternative suggested by ‘Un francophone’)

Dave

Not a real answer, just a personal comment.

I agree with the accepted answer, however I would like to point out that this is very informal. Ending an email with only the word "désolé" is very abrupt.

It’s exactly the same as using only the word "sorry" in english .

In a professionnal context, I would write why I am sorry. For example:

Désolé de ne pas pouvoir être là.

Désolé du dérangement.

…

Furthermore, adding a

Cordialement,

Before the signature cannot be bad.

According to the kind of relation you have with the other person, it might sound a little bit inappropriate. If you answer one of my request so laconically, I am not sure I would recontact you again. But actually I think it is the same in english.

I would consider an email saying

I will not be available the weekend of July 20.

Sorry,

Dave

very rude.

I wouldn’t say it was ‘very rude’, just slightly inappropriate, totally depending on the recipient.

Just restructure your first sentence, remove désolé and it’d be less blunt?