How can you modify Milgram's experiment so that it is ethical and has no potential to harm the participant's mental, emotional and social state?

Despite its horrific (and often misunderstood) outcome, the Milgram experiment really was not that unethical to begin with. I think many people want to think that it was more unethical than it was, simply so that they can place some emotional distance between them and the result. But in fact, Milgram's only unethical actions (by current standards---at the time standards were much looser) were in not getting sufficiently informed consent, overly pressuring experimental participants, and not providing sufficient debriefing. The core structure of the experiment was sound, and the deception itself was not ethically problematic.

It would be quite easy to re-do the experiment in an ethical fashion by fixing these problems.

Actually, this has more or less already been done.

An experiment led by Molly Crockett with Oxford and University College London conducted an experiment very similar to Milgram's, but with much more rigorous informed consent, much weaker pressure on the participants, and much better debriefing. Crockett has also called it the "honest Milgram experiment", because it involves no deception. The participants knew that they were delivering shocks to actual people, and they were. These shocks were much milder and carried very little health risk, but were enough to be painful. Of course, it's impossible to do anything that has zero risk, but the standard used by Institutional Review Boards (IRB) is that it must pose no more risk than daily life, which this study satisfies. (After all, you could be hit by a car on the way to the lab; but that's not an ethical flaw in the experiment.)

What Crockett's team was actually looking for was whether people would really behave according to their own self-interest as standard neoclassical economics predicts. Participants had a series of choices to make, in which they would be paid money either to shock other people or to receive shocks themselves.

The neoclassical prediction is that people would shock others in essentially unlimited amounts for any amount of money, while they would only accept small shocks to themselves for relatively large amounts of money.

What Crockett found was the exact opposite of the neoclassical prediction: People were much more willing to accept shocks themselves than they were to shock other people. Not only were people not selfish, they were highly altruistic---in Crockett's terminology hyperaltruistic, meaning that they seemed to value other people more than they valued themselves.

A lot of subsequent work has been involved in figuring out why this would be true, or what exactly is going on there, because clearly people are not actually hyperaltruistic in general---or we'd all sell all of our possessions and donate the money to UNICEF. A number of new hypotheses have emerged about how human beings make moral value judgments, and this is now an exciting field of research (that I hope to be directly involved in soon, once I start my PhD this fall).