I never did see a response from the OP on what size is washing machine drain is. Also important is the height of the standpipe trap, and the distance that the trap outlet is from the tee where it vents.
I don't know about other codes, but our oregon code is quite specific on this. Trap roughed in between 6" and 18" from the floor, a minimum of 4" of pipe on the trap arm (section between trap and vent), and the standpipe must be taller than 18" but no more than 30".
I also would be concerned about using a gate valve to throttle the discharge.
is that the main drainage stack for your entire house? If it is, i'd run around and turn on your tubs/sinks, let them run for a bit, and then flush all of your toilets at one time with sinks/tubs still running, and see if you get a backup out of the laundry standpipe. This would probably tell you if you have a backed up main.