Is it possible to block connections to a web service (server) from outside its domain?
Certainly. Set your web server's access control lists to drop connections from outside of your IP range. Alternately, install a firewall. That's very straightforward, but I suspect you mean something else by "outside its domain?"
From your description, you seem to be really asking whether you verify that you're only talking to your own client application. As a general rule, no. You can authenticate users. That's easy. If the user isn't logged in and authorized to use your service, you don't forward requests to Twitter. But you can't authenticate applications.
If you're going to accept any user who shows up, you can't stop them from using whatever client they want. There is no way to ensure that it is your unmodified client if you've allowed it to be run on their machine. They can always modify it, and they can always send you arbitrary traffic from other programs and you can't tell the difference. On the network, bytes are bytes.
It's not all hopeless; there are things you can do. See https://stackoverflow.com/a/9183066/97337 for another version of this question, and links to several other versions of the question. (They're not exactly duplicates in how they're asked, but they all wind up being basically the same answer.)