Gmail addresses aren't recognized when sent from Apple mail

Hmmm,

Safe Boot, (holding Shift key down at startup), does the problem occur in Safe Mode? Could take 10 minutes.

Safe mode attempts to repair Disks & clears lots of caches & loads safe Drivers, & prevents loading of 3rd party extensions, so if Safe Mode works try again in regular boot.

To find out if it's system wide or user specific, try this...

Open System Preferences>Users & Groups, unlock the lock, click on the little plus icon, make a new admin account, log out & into the new account.

Does it work in the new account?

To see if it's your provider, What's my ip...

http://www.whatismyipaddress.com/

Start with these three, check SpamCop or SpamHaus to see if your IP is there...

http://spamcop.net/bl.shtml

http://www.spamhaus.org/lookup.lasso

http://www.spamhaus.org/zen/

Sometimes an ISP will bloc a whole other ISP too, if it's the source of too much SPAM.

Do you mean they don't send or what?

It may just need an App Password...

https://support.google.com/mail/answer/185833?hl=en-GB

Can you send from your WebMail in a browser?

Are you running any VPN, Anti-Virus, or Cleaning apps?

Thanks, by user specific. I didn't mean the accounts you're sending to, but your logged in user account, can you try creating a new user & logging out then into that account for a test?

OK, good luck my friend.