I have questions about "one".
Is the following sentences grammatically correct?
He has three brown rabbits and I have two white ones.
He has three brown rabbits and I have two white.
He has three brown rabbits and I have two white.
To complement @James K's fine answer, there is a way to say "white" without repeating the noun "rabbits" or the pronoun "ones" and that's by using a "that" clause.
- He has three brown rabbits and I have two white ones.
- He has three brown rabbits and I have two [rabbits] that are white.
The first option is still preferable if only because it's slightly shorter.
The version with "ones" is much better. You may sometimes hear something like your example two. That would be more likely with words that can function as both nouns and adjectives. In that case you would probably use the plural form.
For example "a grey" can mean "a horse with white hair". So you could well say
I have three brown horses and he has two greys.
Because the word "grey" has a particular meaning as a noun.
But there is no word "white" that means "a white rabbit" in common use. So your in your example you would say
... two white ones.
The word "ones" in this example is a noun (or a pronoun) and not a numeral.
In some cases the "ones" can be omitted, it is more likely with two contrasting adjectives: "Your right hand is clean, and so is your left."