How can I restore a deleted Apple ID? I am unable to use my Apple watch without my old Apple ID.

As it says in:

How to delete your Apple ID account - Apple Support

"When your Apple ID account is deleted, your account details and the data associated with your Apple ID are permanently deleted from Apple’s servers. You can't sign in to your account, and you won’t be able to access any data, content, or services that were associated with your Apple ID."

If you are the ORIGINAL owner, then see the 'Start a support request' section in:

How to remove Activation Lock - Apple Support

You can ask apple to remove the activation lock online on this page;

Online Activation Lock Removal Tool.

Or this;

If you can't remember/gain access to the apple id that paired the watch to the phone.Take the watch and it's receipt along with some photo id and any other proof of ownership to an Apple Store and request that they remove the Activation Lock for you.