I mentioned my issue in the title. Doing gallops (0-000-000-000-00, or 000-000-000-000) or continuous 0000 riffs on the low E for some time makes my right arm and wrist eventually tired. I always push myself and I realize I may be pushing myself before my muscles have built up for these licks.

Do I just have to keep at it and build up my muscle?
Smaller movements, less tension, push yourself.

It's impossible to tell if it's a stamina issue or not if without seeing you play.
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Just keep at it you will build up endurance over time.
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Make sure you're using only your wrist for the motion and not tensing/using your whole forearm. You sound like someone that only recently learned galloping. Once you have the basic technique down, it's just a matter of tightening up your picking motions and minimizing tension (some tension will always be there). Over a long period of time (read: not in just a few weeks), you will increase your stamina.

Even though I've been doing gallops for a LONG time, I still get tired if I don't warm up first and break right in to songs like Slayer's Raining Blood, etc. It greatly helps (and is probably athletically appropriate) to warm up before asking that much of your body.
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As Zaph said. Tighten up your picking movements and don't allow the pick to travel any further than it needs to. Relax your wrist, don't tense up too much. Sounds like you might be tensing your wrist.
You can definitely improve stamina, but metal rhythm WILL tire out your arm, eventually, at least chug heavy stuff