It really depends on both the context and the sentence. In writing (especially formal writing), it does look very bad to write incomplete sentences. In casual writing (like an email to a friend), incomplete sentences are quite common. You've probably noticed that a lot of people write incomplete sentences here on Italki. In some cases, using an incomplete sentence can make your writing sound more like a casual conversation, which can make it seem more friendly. However, I think it's hard for English-learners to know what kinds of partial sentences sound okay, and which ones just sound like bad English. It's often safer to say the complete sentence until you're familiar with what kinds of partial sentences sound normal in conversation, and what kinds of partial sentences a native speaker would never say. You can probably pick up a lot of natural-sounding incomplete sentences from watching English-language movies or TV shows. By the way, in your example, "Just watching TV" sounds strange to me not because it's incomplete, but because of the tense (I'd probably say "Just watched some TV").

It is common for native English speakers to not use complete sentences when they speak. It is informal.

In your example, to keep the tenses aligned, it would be more appropriate to say: B: Just watched tv. To answer your question, no, that is not true. I don't think it has any correlation to English fluency. However, depending on the situation you may appear overly familiar or too casual. For example, if you were talking to your work boss I would stick to using full sentences. But among friends or family, it's perfectly acceptable to use incomplete sentences or phrases.

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