Not a chance. Why would Galad fight Gawyn when to stop him he can simply tell Gawyn what he needs to know to stop wanting to avenge Morgase? Not only Galad knows Rand is blameless, but the wrongs done to Morgase both Rand and Galad have avenged.

Unless things go totally sour between Gawyn and Egwene, which is unlikely, then Gawyn won't ever betray his promise to Egwene by attacking Rand and he can't do anything to Rand before Egwene tells him she gives up trying to find evidence Rand didn't kill his mother - and Egwene is no fool and will not do that unless she gets proof she was wrong and Rand killed Morgase. Gawyn can be rash at times, but after TGS and Egwene's reaction to his disobedience and Bryne's shaming speech to him, would he be stupid and immature enough to throw his relationships to both Egwene and Elayne away, and his duty to them? As women, neither Egwene nor Elayne would ever forgive him, and as Amyrlin Seat and Queen of Andor they just can't forgive and act like that. The Amyrlin's warder or First Prince of Andor attacking TDR as TG begins would be a complete nightmare for both Elayne's and Egwene's leadership. When you serve people in such positions, you can't do anything to undermine their authority. Egwene would lose all credibility to run the Tower if she can't even handle a warder (which is precisely why Egwene didn't bound Gawyn until he understands that and commits himself to that).

It seems fairly unlikely at this point that Galad and Gawyn will be reunited without Elayne and Morgase being nearby - in the same city anyway.

The way things look now, both Galad and Morgase will come with Perrin to the "big reunion" at the end of TOM. Depending on the circumstances, Gawyn could even learn the story Morgase has returned long before he ever comes close to Rand.

And finally, it's most unlikely Gawyn and Rand meet in any private situation before the whole issue has been totally defused. Rand would most likely never agree to duel Gawyn anyway (he's bind him until he calms down, or something like that), and if he did nonetheless, the two guys might beat the Guiness Records of being bound by the highest numbers of flows of air, and the greatest number of women mutterings about the stupidity of men even heard in a single room.

At this point, the odds Gawyn ever goes forward with trying to kill Rand are extremely low. He had his chance in LOC, but he chose to abide by his promise to Egwene. I'll be quite surprised he gets another chance, not before he's already been shamed for wanting to kill a man he should honour for avenging his mother for what a Forsaken did to her, and saving his nation from falling to the Shadow. The later shennanigans about wanting to give Elayne thrones she can claim legitimately aside, Rand is blameless in Andor, and even quite commendable for what he did to keep it together (though his miscalculated PR about giving the throne to Elayne, then his sudden departure, left the door wide open for the war of succession. It wasn't the only contributing factor, however. Elaida started it, Morgase was too weak to stop her, then Rahvin used the crisis and finally Moridin interfered with some of the players. Rand merely added the cherry on the sundae by weakening enough Elayne's already weak position...). It's not his fault Morgase had already fled when he dealt with Rahvin. Has she not gone, Rand would not have ruled in Andor for even a minute. He'd have killed Ravhin, met with Morgase and left.