“right” vs “correct”
I've had this question for a long time. Which sentence is grammatically correct?
Is this the right way?
Is this the correct way?
I think this pretty much answers your question.
They would be used in different circumstances. For example, if you were lost, you'd be more likely to ask 'Is this the right way?' rather than 'Is this the correct way?' But the answer to your question is that both are grammatical.
They're both grammatically correct. Indeed, they're both grammatically identical: "Is this the [adjective] way?"
They're also both commonly used.
They are close in meaning, and there are circumstances where you could use either.
"Correct" has a nuance of precision that would make it more appropriate when you are looking for an exact answer, (rather than if there are several right ways, and you just want to know you have one of them). Even here though, you could still use either.