Let's get over this ego-baiting and just get to the point:
The leader is not the "Healer" or the "Tank", it is the person behind it. If you are not a leader, it does not matter if you are a tank or a healer, you will not lead unless that is your personality type. Admittadly, I notice more people who are leaders tend to gravitate towards tanks - because of the nature of the position. But that is my own personal bias/view.
Healers do not set the tempo, nor do the tanks. A good tank listens to what their healers is comfortable with. A good healer will have faith in the tank and work to keep them up while they rotate their cooldowns. One can not fundamentally function without the other in any form of semi-challenging content.
Now, as for the OP? They did it completely wrong. They should not completely ignore people and attempt to pull only the mobs they want without so much as a considerate word. However, the words given to them were confrontational to begin with. "No skipping anything" is just plain rude, and very few people will respond to a direct disrespectful order. Answering back politely would've been the ideal - "I'm sorry, I don't mind killing mobs on the way, but I don't want to clear side rooms. Is that alright?" If they disagree... well, shit's bound to hit the fan at some point, I'm sure.
But if a DPS/Healer ever pulls, they can feel free to tank it. You don't really need Heals in the first part of DD if you use the crystals. Bloodbath/Self-sustaining stoneskins will more than keep you alive if you fight within the light of the crystals. That is where my limits begin to be hit - your DPS/Healer should -NEVER- pull. Ever. It is only a tactic of spite and grief when they do so, and I will not, under any circumstance, tank a mob that is pulled by a DPS. Even if it leads to a wipe.
Kick them next time and move on. And if they cancel your vote, then you can cancel theirs just as well. Your friend leaving only hurt you and your friend and gave them exactly their way.