Can I use "It's cold" instead of "It's cold today" just to mean "The weather is cold today"? Once I used this "It's cold" but my friend argued that it was meaningless to say without using the word "today".
Yes, you can say "it's cold." The subject, "it," can reasonably be inferred to be referring to the weather in this case.
"Today" doesn't actually do anything to help specify what "it" is referring to - if you couldn't infer it's about the weather, then you would have to explicitly say "The weather is cold."
Do keep in mind that if you just took a sip of a Slurpee, "it" would most likely refer to the drink. So maybe you do need to be more specific. Context does matter.
What "today" does specify is whether that is a temporary or permanent trait of the weather. I think, though, it would be a trite observation, if you lived somewhere that is cold year round, to say "it's cold" and to be referring to just in general rather than today specifically.
The is in "It is cold" conveys that the reference is to "now," "at present," which should be enough.
(@Lawrence has already said as much in his comment at OP.)
It depends on the context. I could say "It's cold" in different situations and you would probably know what I meant.
- Having just taken a sip from the cup of tea you handed to me.
- Having just climbed into a swimming pool.
- Having just walked in to a room.