See here.

Solid wood will warp like crazy unless you carefully flip the grain alignments correctly.

As someone who has been doing woodworking for decades, I'm afraid that this proposed project sounds like a disaster in the making: You will definitely get warping no matter how skilfull you are with this construction method. There's a reason doors are made with rails and styles and cores that aren't prone to warping. I understand you want a simple-looking door, and they're available, but you are not going to be able to build one yourself with the approach you describe. It would be a shame to invest all the effort into your build only to have it cause all kinds of problems after it's installed.

And the proposed wood species is ________?

I am curious as to how much that table weighs and how it looks and how does it clean when sometning spills. How would you insulate the door and 2x2 lumber is not usually perfect just like 2x4 I would need to see the table and what you are actually planning for the door but the advice above seems right to me,

There is a reason that slab doors use veneers , over stable, foam insulated, composites. R value need really is a thing anywhere there is weather. Stability in an element that keeps out the weather is important as well. Most custom wood doors will use super kiln dried wood, with very very low moisture content. They don't get good R value, but the door doesn't turn into a potato chip.

It seems your primary goal is a 'look' of horizontal 2x softwood. If I were going for that I'd use 1x2s laminated on top of each other with an offset pattern. I'd use a two-part epoxy like West available from boat stores. I'd pay attention to the end grain for the assembly. You can't use epoxy as a finish. UV breaks it down. I'd use 3 ball-bearing hinges.

Can you, yes. Should you, no. But an exterior door and apply the wood to the face as a laminated 1/2" thick piece. Certainly be more stable. Use a marine epoxy and I would use a marine varnish finish as well.

Yes you can. They've built uilt doors out of solid wood for quite a while. I've made doors before. With solid wood panels. Using style n rails you can help with wood movement. I've never heard of super dried wood.

https://en.wikipedia.org/wiki/Thermally_modified_wood

https://arborwoodco.com/

" The guides I see online and in books about how to make a door involve the rails and the stiles and the panels and just do not appeal to me at all. "

@aklogcabin, they want to build it with NO stiles and rails.

Yes I understand the question. But it also seems that folks said no to all handmade solid wood doors. I believe that I would probably have tried if a customer wanted one. Largest glue up of a panel per say would have been a hard maple cutting board for a kitchen island. 2-1/4" thick, 6x7'. A skilled woodworker should be able to make them. The OP asked if solid doors can be built. I said that they can

Points taken. I guess I knew I knew all that but I was hoping there'd be some way around it. I do happen to have a bunch of wood-look vinyl plank, floor leftovers, that could be affixed to the door and give roughly the same look, but I just don't like things that look or try to look like other things. To me if it's not real it's not worth doing.

@Patrica Colwell Consulting I don't know how much the table weighs but considering it contains 24 linear feet of 6x6 solid wood beams my guess is pretty heavy. It was designed to be semi-immobile, to be a kitchen island in all but certain legal definitions.

Timber plank doors are nothing new, nor is making them work really difficult to understand, it is just a lot of work. If you look at the oldest timber plank doors they would strap two or three pieces of steel to the door and use rivets to keep the door from cupping and to limit the movement. Today, if you are using a species with a lot of wood movement, then you typically drill it and use all thread plus epoxy to stop wood movement.

Teak and Mahogany solid plank doors are not uncommon and make fine doors. White Oak is also fairly common but is going to require some know-how. If this is what you really want, then yes... it can be done.

However, this is not a DIY friendly project. If you want to do this for the pride of crafting your own timber door then I would plan on making three or four doors to get one right even if you research it extensively before you start. If you are doing it to save money, you are better off getting a second job as it will take far less time.

I don't see wood look vinyl plank working either.

ETA: If you just want the look then take a couple of sheets of marine grade plywood, cut the 2x2's into strips (1/8" to 1/4" thick) and glue them to the plywood. Use a full 2x2 on each side.