"Anyone tries to deny his destiny" is correct; although, this sentence is kinda strange in the sense that I don't know where you would ever need to use it or why / where you would say it on its own like this. I could see it being used more as a subordinate clause in a sentence. I will give you an example of this below. By the way, I think most Americans would say "deny their destiny" in this sentence to mean a person in general. That way there is no gender associated with the pronoun (male / female). I know this looks strange because "their" is technically plural but this is what we do in English. You will see "their" in my example sentence.
Also, how about using a pronoun like "everyone" instead? "Everyone tries to deny their own destiny." is an example. This sentence sounds natural on its own. Or if you really want to use the pronoun "anyone" you could also practice with these sentences:
1. If anyone tries to take my book, tell them to leave it alone.
2. If anyone tries to steal your purse, spray them with mace and then call the police.
3. If anyone fries some zucchini, can you put some aside for me to have as a snack later?
4. I get very angry if anyone tries to eat my lunch on me.
5. I get very annoyed whenever anyone cries during a movie at the movie theater.
6. I get very frustrated whenever anyone tries to deny their own destiny.
(Here is your example used as a subordinate clause) :)