To answer your question directly, no.
Refresh rate is simply the frames per second the monitor is capable of actually displaying and has no bearing on screen tearing.
However, higher frame rates will result in less tearing in most games and having a monitor that is capable of displaying those higher frames helps achieve that. But it does not remove screen tearing all together, and playing at lower frame rates will result in worse screen tearing.
The only way to fix screen tearing all together is to enable some kind of sync. Whether it be through hardware with FreeSync/G-Sync, or through software using V-Sync.
But to make things smoother without any syncing, you need higher frame rates which would require you to lower graphic settings to achieve.