A trusted certificate is signed by a CA's private key. What makes it a CA is that it
a) has special markings that denote it's a CA (well, sort of), and
b) it's public key is stored in your trust store meaning your computer trusts it and it's children.
The certificate you have is intended for a specific purpose, and has it's own special markings that denote what it's used for. Client certificates also have their own markings, etc.
Now, you can use your private key to sign other certificates in the technical sense, but nobody will trust them because your certificate doesn't have the special markings that denote it's allowed to do that.
So yes, you do need your own CA in order to use client certificates the way you're wanting to use them. That said, every service that needs to verify one of those certificates would need a copy of the CA's public key in their trust store. Of course, if you did have the power to sign such certificates that were trusted, you would have an immense amount of power at your disposal because you could create trusted certificates for services that aren't rightfully yours.