Rosary Clinic in Springs would see you; they have an emergency department and it's probably 10 minutes [tops'> drive from where you live.
Just front up at the reception area and you will be seen to almost immediately. Emergency cases aren't common there so they will bump you to the top of the queue.
It IS a private clinic though, so you need private health insurance to be seen. ;)
Probably Medcare opp Safa park and JESS jumeirah. Alternatively you could go to Welcare in KV. For simple stiches both of these will be more than acceptable.
Where do you live? If you're new Dubai end Cedars International at Jebel Ali port is easy and fast. Taken my two little ones there, stitches both times. We were in and out very quickly. First time they were fab, second time the doc seemed nervous (head stitches) but amazing scars both times, quick service..... It's not pretty and shiny but good! Otherwise City Hospital or American Hospital?
I've no experience, but there was a thread here recently which definitely voted for Rashid for that kind of emergency (well, trauma after car-accidents, in fact, but rated v good for emergency treatment).
The 3yr old daughter of my visitors has sliced her thumb open pretty deep (its not bleeding now) but will likely need stitching - whats the quickest/easiest place to take her for this?
Was thinking Rashid trauma centre but not keen on ay long waits.