'Dw i'n gwylio'r rygbi' would be also translated into English as 'I'm watching rugby' (without the definite article) and this would be the most common expression in British English.
The phrase 'I am watching THE sport' is extremely common in English in Wales as it is a direct translation from Welsh.
All sports should be 'The sport' in Welsh.
As an English English speaker, I too would say " I'm watching the football" or "the rugby" and I think that's standard UK usage.
I wouldn't say "I'm playing the football" or "I like the football" though.
As a native English speaker, I would say that "I'm watching the rugby" is synonymous with "go away, shut up, leave me alone". I would only say "I'm watching rugby" if I'm not sure which teams are playing, or if it's an old game that I've seen before..