To be clear, although there are rules in the English language regarding grammar and proper English, as with most everything, this matter is completely affected by a person's perception. Even rules are affected by perception. The reason why this is important is that there are more than just two dialects of the English language, and not every dialect will agree upon the same rules. This is why there are dialects of most languages in the first place. Someone thought it sounded good, used it, and after enough people used it regularly, it became a dialect, regardless if it is official or not.
I may feel that it is not correct, but another dialect may say different. A lot of the answers here mix up the rules of grammar and do not particularly stay aligned with the sentence in question, as stated by Mari-Lou A. Yes, it is important to have a "rule" or standard (both mean the same in this case) that can be applied along a range of sentences. That is what makes it a rule or standard, but when you compare adjectives versus nouns and say that one is more correct than the other, you lean away from the original question by saying that something does not function properly, but rather than defining your reasons with semantics, you demonstrate it with another example that utilizes different semantics, which explains why someone else perceives that reason as incorrect. You may be able to reason an example of complete difference logically, but if the semantics differ too much, then it is difficult to understand the relation, and even more difficult to follow that logic to the point of agreeance.
This does help to deduce the answer, but I feel that the answer is quite obvious. It's a matter of opinion, based on the audience in which it is meant for. If you want all native English speakers to believe it is proper and correct, then you have your work cut out for you, but if you're just looking for a majority's acceptance, then stick to saying "I hate the color red".
The point is to understand what makes a complete sentence, and thereby you are able to deduce the acceptable answer here. A complete noun and verb make a complete sentence, and this is generally the most acceptable way to convey proper, acceptable English.
"I hate red color." - Granted, a native English speaker would most likely say that the words "red" and "color" are switched around, but that does not mean it is grammatically incorrect, although it would be missing an article if that was the case, hence making it, in fact, grammatically incorrect. To be precise, it isn't standard (there is a difference between what is standard and what is grammatically correct), and because it isn't standard, nearly all native English speakers would correct you by saying it is switched around.
To further my point -
It is proper to say, "I hate the color red" rather than "I hate red color", specifically, because of the order of the nouns. In the version I presented, "the color red", "color" is a noun used as an adjective describing the noun "red". If you say, "I hate the red color", you are still using a noun as adjective "red" to describe the noun "color". Either way, depending on the context it is used in, you can also be stating a compound noun, because the latter noun "color" or "red" isn't necessary in that case, but this sentence on its own does not convey a complete thought, because it is implied that you are meaning the color of something that isn't stated in that sentence. In other words, if you say, "I hate the red [color]", it is obvious that there is more to this thought that isn't stated.
As Casey LeClair said when saying "I hate the color red", you aren't specifying its medium or location in relation to a shade of red. Therefore, it would be accepted as hating only red generally. But as JohnGH said, it calls for an article in this case, because the subject is singular, while depending on the plural, it is not always necessary to add an article. This is the generally accepted rule.
I agree that when saying "red color", red is an adjective, but can be a noun in a compound noun of "red color", because as said by moonring and J.R., it depends on the context, and there are exceptions to rules.
The basis of the answer here is that it depends. It depends on the person perceiving the statement. Honestly, you might never know which to use, so use your better judgment and sound it out both ways. Native or non-native, either way can go. This doesn't mean always, but don't take it for granted. Grammar nazis are everywhere.