Simon Yates won the challenging Bastille Day stage, distancing Healy on the final climb to claim his third career Tour stage victory.
Healy finished third on the stage, gaining significant time on Tadej Pogacar and Jonas Vingegaard, putting him 29 seconds ahead of Pogacar going into the rest day.
The 165km stage through the Massif Central featured eight categorised climbs and 4,500 metres of elevation, proving ideal for a successful breakaway.
Remco Evenepoel lost six seconds to Pogacar in the final, now trailing the world champion by a minute and sitting 16 seconds ahead of Jonas Vingegaard.