Constitutionally, no, the right referenced by that phrase in, e.g., the Miranda warnings is the 5th Amendment right against self-incrimination (“No person... shall be compelled in any criminal case to be a witness against himself”).

The belief that a more general “right to remain silent” exists may be a moral belief of some people, but, it is not a legal right, and judges can and will conduct questioning to assess if the legal right applies.