I think your usage, *"I hear it a lot while watching cartoons ..."*, is correct. The word *cartoon* is a countable noun. A singular countable noun will need an article (i.e. *a cartoon* or *the cartoon*) or a determiner (e.g. *that cartoon*). Thus, we can say ***watching a cartoon*** or ***watching cartoons***, but not ***watching cartoon***.

However, you observation made me wonder if "watch cartoon" had become a phrase (as "declare war" did) already.

So I looked into some of the examples returned by Google Ngram, and found that all the *"watching cartoon"* results (as far as I reviewed) are not a valid noun phrase. For example, *All your little friends were watching cartoon shows ...*, *... watching cartoon turtles on TV doing ...*, *... that children watching cartoon shows for a four-hour period ...*. In all results, the *cartoon* was used as an adjective.

With that, I think I can conclude that saying ***watching cartoon*** is incorrect.