If the space selected for the player to land after passing through an End Gateway contains Iron Bars (maybe other transparent blocks too, but I had iron bars because they're dragon-proof), the player falls through the floor, which can lead to falling into the void.
Easiest way to reproduce: defeat the dragon, which will cause a gateway to be created at -56,75,77 (quite possibly over nothing but void). Build a small platform around the gateway, say 7x7 at Y=74. Place iron bars around the perimeter of the platform. Ender pearl through the gateway, and then ender pearl back.
When you return you will fall through the platform that you built.
Further testing confirmed that the gateway will attempt to place you as far northwest as possible, up to 5 blocks in each direction away from the gateway, depending on where it can find a solid block to put you on. If there is a solid block at -61,74,72, that's where you'll end up. If that block has an iron bar on top of it, you'll fall through.
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This still happens in 16w15b.

Is this still an issue in the most recent versions (currently that is 1.10.2, or 16w43a) of Minecraft? If so, please update the affected versions and help us keeping this ticket updated from time to time. If you are the owner/reporter of this ticket, you can modify the affected version(s) yourself.

Still happens in 16w43a.

Can't reproduce the issue in 1.11.2

I can still reproduce in 1.11.2. Would it be helpful for me to provide a world download?

Yes, it would

Ender pearling through the gateway in the attached world causes the issue in both directions. I'll test now to see if it still happens in 17w13b.
Edit: it does.

Seems to only push you through the floor when the chunk containing the destination is not loaded before entering the gateway.
Edit: Also found to work with glass panes, stairs (not slabs), cobble walls, fences, fence gates, Shulker Boxes, chests and carpet, as well as many other similar blocks.

Still happens in 17w50a

Teleporting inside non-solid blocks occurs in 1.14.4 and 19w39a.
Slabs don't seem to be an issue, but they technically are, it's just your player will pop up above the slab automatically (if it is placed on the bottom segment only) so it doesn't cause as many problems.
I can't seem to reproduce in 1.16.5. The player is teleported inside of the iron bar (in the attached world), but does not fall through the blocks.

Please check if that's still an issue for you in the latest 1.16.5 stable release or the latest 1.17 development snapshot.