As shown in the attached pics, I've noticed that right after the first true leaves of my melon seedlings became established, about half of my seedlings have started to bend over, leaning to one side.

It doesn't look like they're necessarily wilting and the stems appear to be about as thick as the plants that are completely upright. They're all getting about the same amount of light (16 hrs under LED grow light) but some are further from the "optimal" center zone, so maybe they're just stretching for light?

I'm just wondering if the leaning tendency is the plant trying to tell me something (i.e. too much water etc.) or if this is pretty standard given how the plant naturally grows...I know melons can be sensitive with water levels / prone to damping off so I'm hoping it's not the start of that. I amended the soil with diatomaceous earth and worm castings which significantly improved water retention over time, to the point where I'm a little nervous that it's creating conditions for root rot.

They probably lean toward the light.

^^^What Tammy Sue, the Melonator, said. They look to be searching for their energy source.

I see a heat mat beneath some of the containers. Are you still using it or just haven't removed it yet? I would think you wouldn't need it now.

Something you might want to do to enhance your LED grow lights performance is to hang some mylar where you have the blue tarp hanging (hang it over the tarp if you want to, it's lightweight). It will reflect a lot of the wasted light coming from the grow-lights back to the plants. Look for cheap "emergency blankets" in the camping department at Walmart or some other department or sporting goods store, they work well. Also, rigid foam insulation with foil backing (even with printing on the foil) works well and can actually create a semi-insulated pool of air around the plants...it's easy to stand/prop up if you get 5/8" or thicker sheets....3/4" works nicely. Even just a solid white sheet of "something" to reflect back that lost light will work better than a dark surface.

Hopefully before long you'll get to move them on out to the garden where they can reach for the sky!!!

I left the heath mat on for just a couple weeks after germination - looking back I would probably have turned that off even sooner as I think the extra heat was causing these to almost grow too fast, becoming a little leggy even with what I expected was sufficient light, with the LED as close as safely possible.

Completely agree, I think a reflective cover would be a game changer and will probably implement this in the next grow.

My cantaloupes lean, they are ground vines, they don't usually stand up like tomatoes and peppers, without staking my cucumbers also lean over as soon as they get growing. They lay along the ground because their leaves are too heavy to hold in the air with such lightweight main stem. Hales Best Cantaloupe.