Is it common for LO (at stage 6) to keep eyes closed all day even while talking to me? He says he can open them and he does for split seconds.

People with dementia have a difficult time processing information, the brain just does not work properly.
By keeping his eyes closed he can block out other senses that compete with the voice he is listening to.
His body may also be slowly shutting down.
I would suggest that if he is going to a Skilled Nursing facility that you also get a Hospice referral. Hospice will be another set of eyes on him, They will make sure he is comfortable. There is a very good possibility that he will not be able to fully participate in rehab. I hope I am wrong as this is difficult to hear.

Is PT working with him in the hospital? did he have some ability to support his own weight prior to the fall that led to this hospitalization? can he remember instructions such as "call me when you need to stand up, please don't do it alone"?
If any of these are a 'no' answer, then SNF for rehab (if possible) and then residential care are likely the appropriate level of care for him.
You can also arrange for a hospice agency to provide services to supplement what the SNF provides, and provides support for him and you at this stage of his dementia. Hospice doesnt mean he is actively dying....but he is getting weaker with every fall...and now closing eyes to block out the world around him.....
And try not to feel badly about placement - if you fall or have serious health problems develop, that is where he will be placed. Better you be part of the process of choosing, and adjusting to SNF.
Good luck...this is a tough part of the journey.

How about asking his doctors if over-the-counter Theratears for dry eyes is allowed? I use them a couple of times a day, and they keep my eyes lubricated very well. They're especially helpful at night. The version that comes in packs of single-use ampules has no preservative, and that's what my ophthalmologist recommends.

My mother did this on occasion. And when she did it, she was more lucid. Its been discussed before on the forum and the thought is by closing your eyes you are blocking out all outside stimuli that can be overwhelming. Might be better for him he does this.

Please, don't feel guilty about placing him. This is the best for him and you.

I am glad that you recognize that you can't handle it anymore. The best way is to know that you can't. As he may now be SNF care for some time this is the BEST TIME to reach out as soon as possible to their social workers and discharge planners to make it clear that hubby cannot return home any more. The sooner they know the better.

You say that he has had a brain bleed and that this eyes-closed behavior is new since then if I understand you right? Then you will need to wait a while to see if this automatically clears up or not. Does he do this when walking with PT also? Or only when at rest in bed. Our eyes can get so dry in some environments it becomes almost second nature to close them, then almost habitual, but this seems to have happened secondarily to the bleed. Let PT and OT if they work with him at SNF know that this is new for him; that will help with his assessment.

I surely do wish you the best and hope you will update us. Remember, the Social Workers and Discharge planners are your new best friends. You may be/should be included in an initial care plan assessment; ask to be included. You can let them know then and there that you cannot take hubby home, even with some improvement. They will likely respond "Well, we can help make that work, but right now we will do intake assessment and hope for improvement". Let them know that even with improvement, you are now at the end of what you can endure.