He isn't on Hulk Hogan's side. He's on his side, and he won. Making this about the particulars of Hogan's case is deliberately eliding the fact that Thiel didn't give a shit about Hogan's case other than that it could bankrupt Gawker.
1. You have a moral obligation to litigate moral wrongs, whether or not they happened to you. Since rich people have more money and hence more things are litigable to them, they have a moral obligation to litigate more things. In this view, Thiel obviously did the wrong thing: he avoided litigating moral wrongs for years while waiting for the right case to come up strictly about Gawker. He failed to act ethically, under this model.
2. It is morally permissible to litigate moral or even legal wrongs that were not committed to you. It being permissible means it's a supererogatory feature of your moral life, in that you don't have to do it. It goes beyond the call of duty to sue Gawker for what they did to Hogan. But without a moral obligation to do so, you can't say that it could ever override your other obligations to behave ethically. Retribution, which this clearly was, is not an obviously ethical motive that would mean we should wholesale excuse what Thiel did—even though it may have been a good act to punish Gawker for Hogan, it was morally wrong to use the legal system strictly for retribution.
3. It is your moral obligation to litigate all litigable acts. This is absurd, prima facie: all our money and time should go towards litigation? Nah.
In other words, yeah, the motives matter. Thiel went above and beyond the call of duty and acted unethically, in an especially egregious and self-serving way available only to billionaires, along the way.
"Moral obligations" are made-up. Obligated by whom? What happens if someone doesn't fulfill their moral obligation? Nothing, right? Maybe someone writes a comment on the Internet saying they should be ashamed, but that's pretty much it.
Feel free to pontificate all you like about the morality of it. That won't change the fact that it did happen and will happen again unless a force powerful enough to stop it chooses to do so.
Ultimately, the whole point of having a justice system is to sort out these questions. Anybody can sue anyone else for any reason. We depend on the courts to sort out the ones that are valid from the ones that are invalid. In this case, the courts deemed this a valid lawsuit. If you disagree with the validity of it or feel the methods by which the courts make these judgments is flawed, then that's one thing, but the rules of the system allow any lawsuit to be brought for any reason.
In any event "retribution" is an overly simplistic way of looking at what Thiel did. It could also be interpreted as protecting future victims of Gawker.
Suppose someone stabs me, and I live. For whatever reason, they get away and continue to run around stabbing people. If at some point in the future, they attempt to stab someone else and I stop them, killing them in the process, is that killing in retribution or protection? Maybe both. It doesn't really matter so long as we can stop people from getting stabbed.
Gawker went and made a powerful enemy for no good reason. Outing someone who didn't want to be outed helped no one. Thiel used the considerable means at his disposal to ruin Gawker in a way that didn't harm anyone other than the owners of Gawker. Since nobody else really benefited from Gawker's existence it's unsurprising that nobody else really cares if they stop existing.
If Gawker had been exposing powerful people doing something illegal or otherwise harmful instead of just being gay or having affairs, they would have had the justice system and the public on their side and they wouldn't have lost the lawsuit.
And it was cheap for him - around $10mm to bankrupt his mortal enemy. There's no way that other rich folks aren't going to do it as well, especially now that he's framing it as a philanthropic act.