This is not a legal advice, but here in CA, if there was a shut off valve behind the fridge at the time you moved in - it would be the responsibility of the landlord to make sure it was in good working order and didn't leak. If you, the tenant, installed it - it would be yours to deal with. There are a few more possible situations...that might get even more involved.
If your landlord asks you to pay for the damage or part of the damage and you disagree, consider calling a real estate lawyer. If you agree because you love the place, then work a deal with your landlord.
Now about the shut off valve: an unused 20 year old multi-turn valve with a plastic stem can go bad anytime. I've had brand new valves leak at the first use. Even the modern 1/4 turn valves can fail right off the bat. Don't forget, these valves are mass produced and are under pressure 24/7.
Last note: Insurance would have come handy here (unless the insurance company found a reason not to pay).