Do You Wash or Do Laundry

Both words are used – although “laundry” is more specific to clothes (and towels, bedsheets etc). You use the washing machine to do the laundry. You can also “put a wash through the machine”. In context, you can “do the washing” – if the listener knows you’re already talking about clothes.

Is it correct to say Im doing laundry?

“Doing laundry” is very common.

Do the laundry means?

: to wash the dirty clothes, towels, etc. I have to do my laundry today.