Can I use my Apple Watch with my spectrum carrier??

Your profile says you have Apple Watch Series 1 which does not support standalone cellular. That means you must link it through your paired iPhone, and your network doesn’t know or care about the watch providing your iPhone has data available.

To be sure, see https://support.apple.com/HT204507 and check exactly which model you have. Apple Watch is now getting seriously old, and survivors are becoming scarce as batteries age and users upgrade.

If it's a GPS only watch then yes,if it's the cellular watch then no.