Native English speakers only, why not "study"?

They are both correct. They mean slightly different things.

To study means to attend classes, read, take tests, to do what students do in an attempt to learn.

To learn means to gain knowledge. Some people learn by studying. This is how we all learn a second language. But you can learn a language without studying, as small children do when they're growing up.

They are both nonsense. They are both extremely badly expressed. Neither of them say what you mean to say.

1. Who are the "native speakers" to whom it refers ?

It suggests to me that it is English speakers who are doing the learning. Why should an English speaker want to learn English ? (However, knowing the standard of English spoken by most teenagers in England, it would probably be a good idea for someone to teach them to speak properly).

2. Same objection as above. It is suggesting that English-speakers should study the English language. Which is what everyone does who goes to school, or follows a course in English language at a University. But hyow can you STUDY a language which you cannot speak ? You only STUDY English language when you know how to speak it.

What you need to do is to specify exactly what you mean by "native speakers". If you are claiming to TEACH good English, it is a poor advertisement for your ability if you cannot even construct a simple, understandable sentence in basic English.

Try something like "Learn to SPEAK English as well as English speakers do"