the thing is, that you need to sit down and talk about expectations.
This isn't really about teh washing, as you said yourself, it is about who is responsible for the household tasks.
He will be very influenced by his upbringing over whos job is whos, but that doesn't mean he can't change.
You need to sit down and talk, ask about who did what in his house, ask about what he thinks is fair if you are both working, who should do what.
make a list of all the jobs. Don't forget the things he does, things like bins or cutting lawn (I know I am stereotyping) put a note next to each thing about how often it should be done (weekly/monthly/nightly)
One thing he needs to understand is that he is not a child at home, but an equal in a relationship. If you do everthing, you arenot an equal you are his cleaner.
We have a rule, that if I cook dh clears up. But I know others who say their OH is too messy, so they have it that either cook and clear up, and you take turns during the week.
You need to establish what is a reasonable timescale too, is it ok to leave clearing up til the morning, provided it actually gets done?
Then divide the list, let him choose which jobs he is going to do (or take turns choosing a job each) Agree to do thise jobs for 1 month.
Once a job is his responsibility, you then need to back off, let him do it his way and don't comment. At the end of the month you can both assess how it is going. One thing is though, my dh does some things differently to how I would do them. It is our house. As long as it is done, I need to let him do it his way, my way isn't right, just different.