External Regret and Nash Equilibrium
No, no-regret dynamics do not converge to Nash equilibrium in general games, and its not hard to think of examples. 

On the other hand, no regret dynamics do always converge to coarse correlated equilibrium in any game, and no-internal regret dynamics always converge to the set of correlated equilibria. 

They are also known to converge to Nash equilibrium in several special cases of n-player games: for example, separable zero sum games, congestion games, and a few others. 

In general, there are no efficient learning dynamics that converge to Nash equilibrium though. Not only is computing a Nash equilibrium PPAD hard, but even ignoring computation, for any "decoupled learning dynamics" (among which no regret dynamics are a special case) must exchange exponential communication before finding a Nash equilibrium in general games.