I'd like to install a gas dryer by splitting the gas line that currently feeds the gas water heater. Is the solution as simple as splitting the line with a tee fitting/valve where the green arrow points? Any other parts needed?
Instead of teeing the 1/2" line where both appliances would be fed through the same section of 1/2" pipe, you would be better off installing the tee upstream, where the 1" coupling is now. This will ensure adequate gas volume to both appliances and allow you to install independent shut-offs. Yes, you will have to shut off the gas at an upstream location, maybe even all the way back at the meter.
You should take this time to eliminate the reducing bushings (I see both a 1"x3/4" and a 3/4"x1/2") as bushings are not supposed to be used on gas lines inside the home (check with your local permitting authority, but that's my understanding). I would just get rid of the 3/4" stuff completely and use a 1"x1/2"x1/2" tee and install valves with flare thread outlets to attach the flex line directly to. Iron pipe and fittings use a tapered thread, you need to get them tight for a good seal, and use thread tape approved for use on gas pipe.
If you are handy at all, you can do this job yourself (don't listen to naysayers), you are allowed to do this work on your own home in most places. Make sure you test all new fittings and those that were disturbed using "gas leak detection juice" or soapy water (little bubbles=small leak, big bubbles=big leak; NO leak is acceptable).