While there is no immediate need to defeat Doom Gaze (though it's worth doing so if you can since it drops the Bahamut Esper), it may attack while you're flying around on the Falcon so it's best to be prepared. The first thing to note about Doom Gaze is that it is not a random encounter in the usual sense. Rather than being triggered by some amount of movement or random chance, Doom Gaze actually "spawns" invisibly somewhere on the World Map, and if you run into this area with the Falcon, a battle will begin. This spawn location changes every time you board the Falcon, so if you're hunting for Doom Gaze, you'll want to do a sweep that covers the entire map.
When Doom Gaze attacks, it will always open with L.5 Doom. Keep this in mind when flying around to avoid an unfortunate party wipe. Doom Gaze lives up to its name by casting Doom, but can also cast a frighteningly powerful Ice 3 as well as Aero. In every third round, there's a 2-in-3 chance Doom Gaze will run. With 55,555 HP, this may make the battle seem impossible, but don't despair: Doom Gaze doesn't recover HP between encounters. Note that you can also run from these battles if needed.
Doom Gaze is not technically undead but has the usual undead vulnerabilities to Fire and Pearl while absorbing Poison. It also absorbs Ice. Since any given skirmish with Doom Gaze may only last three rounds, you'll want to hit it as hard as you can with your most powerful attacks. It's unlikely to kill the party before running unless you are woefully unprepared, so focus on offense. If you're on the airship, you've likely saved recently, so go all out.
When you finally finish Doom Gaze, which may be quite late in the game if you don't go out of your way to hunt it down, you'll earn the awesome Bahamut Esper. It teaches Flare at ×2, and its Sun Flare summon is a multi-target version of that same spell. It also offers +50% HP at level up, the best such bonus in the game. It's well worth the effort, and a good incentive to look for Doom Gaze once you're confident of your abilities.