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Trump quickly lashed out on Wednesday, dismissing the op-ed as "really a disgrace" and "gutless" and assailing the author and The New York Times for publishing the anonymous opinion piece.
Trump slams damning New York Times op-ed as 'gutless'
Look dismiss up from dictionary, it means
to refuse to consider someone’s idea, opinion etc, because you think it is not serious, true, or important
I think it's similar to "refute".
To add to your question, here is the Merriam-Webster definition of refute:
1 : to prove wrong by argument or evidence : show to be false or erroneous
2 : to deny the truth or accuracy of • refuted the allegations
When you refute something, you say something actively against its claim. Even if it's just to say that it's "wrong," you are making an active statement in opposition to it.
To dismiss something it to not consider it at all. (As cited in your dictionary quote).
If using the second sense of refute, it's possible to both dismiss something and refute it:
No, that's simply not true [refute], and I'm not going to discuss it any further [dismiss].
In the news story you quoted, Trump is dismissing the story but he's not saying it's false, let alone trying to prove that it's false. In other words, he's not refuting it in either sense of the word.
You said in a comment that in the Chinese translation, dismiss means "don't care." In English, it means "don't care to discuss it further."
