Can an officer issue me a speeding ticket based on approximate speed in California?

Yes. I got one  in Oakville  for doing:  "90 +". It really wasn't necessary in this case to clock me. They Must, however,   clock or use Radar to make 45 in a 40 zone stick.. They can also ticket you for going too fast for conditions; the basic speed law. Say, it's snowing,  black ice, SL is 55, you can still get it  in this  case  for doing 55.

If the speed limit is 35, and your approximate speed is 60, yes.

If the speed limit is 65, and your approximate speed is 66, probably not.

Note: Most officers are trained to be very accurate visually.

In addition to the methods stated by Bruce, the officer could time you between two points of known distance and calculate your speed within the accuracy of the timing. Another method is pacing you. If the officer maintains the same distance from your vehicle, they can read your speed off their speedometer, without a direct measurement.

Yes, they can. I can think of several scenarios. 

VASCAR for one. That clocks your average speed over a known distance. 

Or, he could clock a vehicle at 65 in a 55 and observe you pass that vehilce. He may not have your speed, but he knows it is higher than 65. 

Or, your speed could be so fast it is obvious you are over the limit. 

Or, he is doing the speed limit and you pass him. 

I could go on, but I think that makes the point. 

Yes. The officer can also testify, if you contest the ticket, to what he witnessed and the actions he took. Radar readings are not required to get a conviction.

You obviously haven't read any laws so you shouldn't even be asking. Crack open a law book and figure it out for yourself. Might as well read every single law since you're so clueless.

I'm sure he will if you ask nicely.

thats up to them to decide