I just realize why you hate me
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You can, but not in that way.
If you wanted to say that you've come to know only a brief moment ago that the person you're talking to hates you, then the simplest way to change your original sentence to be correct is like this:
I've just realized you hate me.
However, you can use the phrase "just realize," as illustrated by the reply from the other person:
Did you just realize that I hate you?
Alternate version of the same question in past tense:
Have you just realized that I hate you?
I don't think you can use "just realize" outside of a question, though. Someone correct me if I am wrong on this.
It should be noted that despite the fact that the tenses are different, the meanings are virtually identical. The reason is the use of the word "just," which is used as an adverb to mean that it happened very recently. We understand that the realization happened only a second ago in both sentences even though the tenses are different.
The sentence is technically correct. However, it's most likely never going to be used that way. Realize is the present tense of the verb. For that particular sentence, you would need to use the past tense ("I just realized why you hate me") or present progressive with a revision ("I am realizing why you hate me").