Assuming Rhaegar immediately marries Lyanna, what he should do is, firstly, not let his crazy father deal with the fallout. He needs to come up with a plan to deal with the Starks, the Baratheons and the Martells.
The first to react is Brandon Stark. Rhaegar should be there to deal with him. He should greet Brandon unarmed and tell him calmly to put his sword away and to come in and we'll talk about this. Brandon's looking for a duel; he's not going to murder an unarmed man. Offer him bread and salt. He's not going to break guest right. If he refuses to back down, then have him arrested and summon Lord Rickard. We don't have much information about Lord Rickard, but surely he would be easier to deal with than hotheaded Brandon. Once Rhaegar explains to them that he's married Lyanna, the Starks really don't have much of a choice but to accept it for the sake of Lyanna's honour. They still have to kick up a fuss, mind, because Rickard did not authorise the marriage and Lyanna was married against her will (this will always be the Starks' official story, even if she tells them otherwise). Rickard is ambitious, so Rhaegar could exploit this and make sure the Starks benefit from the marriage (e.g. a seat on the council, a guaranteed supply of food in winter when Rhaegar is king).
Once Rhaegar has the Starks on his side, the threat of war pretty much goes away. The Martells will be pissed, but they have no friends, so they're not really a threat. They're still a problem, though. All Rhaegar can really do is assure them (and the rest of the realm) that Aegon is still his heir and make sure that the Martells get whatever the Starks get. This still sets up potential hostility between the Martells and the Starks but the fact that they're at opposite ends of the continent mitigates this somewhat.
That still leaves Robert. I'm not sure if there is any way to appease him, to be honest, but there's not much he can do on his own. Rhaegar could arrange another marriage for Robert (not sure with whom because Rhaenys and Arianne are still only babes at this point) and could take Renly as a squire/hostage, but I think his best bet is to get Jon Arryn to talk Robert down.
And he should publicly apologise to all parties involved.