Yes, you can. . . and receive news articles from various Internet news sources of your choice. You can also initiate and receive calls and texts, or listen to music or audio books.
Just to clarify for the uninitiated...
The newest Apple Watches CAN have independent cellular capability. That is an option. You can purchase them with or without cellular capability.
It is pretty obvious when you are interacting with your Apple Watch. Yes, you can send and receive texts, but that would require either tapping on your watch, writing with your finger on the face of your watch, or speaking to your watch - all of which would be noticed.
As for listening to your watch, that pretty much requires Bluetooth headphones or earbuds. Again, it would be noticed.
While I understand and agree with the banning of electrical devices, including Apple Watches, doing something distracting with them would be obvious. I do think that someone wearing an Apple Watch (or other smart watch) could receive text messages, which would be against the rules and possibly harmful to the proceedings. They should make sure NO Senator has ANY electronic devices.