I recently noticed that my front wheel was "wobbly", so I looked up some videos and adjusted the spokes to get it true. It seems to have worked, as the wheel originally wobbled when I spun it freely, but afterward it spun smoothly. However, now I notice that when I apply the front brake, the wheel is pulled toward the disc brake so far that it almost touches the fork. Did I screw up the wheel by trying to true it? Or is this a brake problem?

I expect what you did was move the entire rim over toward one side while truing the wheel. Unlike a conventional front wheel, disc brake front wheels are "dished" like a rear wheel so that the rim is centered between the locknuts, not between the hub flanges. If you moved the rim over it is no longer centered where it should be. If you don't have a "dishing tool" or don't know how to determine if the rim is in the correct location, a trip to the bike shop is needed.

Another possibility is that you have play at the hub, or the wheel isn't properly tight in the fork. This allows the action of the brake on the disc to move the entire wheel over.

Step one, before anything else, is to see if the rim can be pushed to the side easily. I'm only talking about free play here, not whehter you can flex the wheel sideways against spoke tension.

Otherwise, do as Hillrider said, and check to see where the wheel is "centered" now. If it's off center remove it and remount it backward (disc on opposite side) and see if the rim switches sides. If so, you're wheel is not dished (centered on the hub) properly, and needs the spokes reworked to move the rim over.

With some disc brakes the caliper only has one moving pad, while the other is stationary. With these you need to adjust the caliper so the disc runs as close as possible to the stationary pad, otherwise it can push the wheel out of true every time you brake.

If the hub bearings are properly adjusted, it won't move. Yes the disc flexes but the hub doesn't move, so therefore the wheel doesn't move either.