Converting PWM to analog voltage
I used a RC filter with a time period of 3.9ms. The PWM output has a
  period of 2ms.


You need a longer time period for your filter I would suggest. I would also suggest that you simulate this to see what the output looks like and notice that some artefacts of the PWM signal are still present on the DC level: -



The above uses your numbers and yes, adding a load will alter the filter characteristics and give different results (more ripple voltage). If you have decent bipolar supplies you should be able to use an op-amp but remember, an op-amp can only usually supply loads of up to 20 mA. If you are looking for a more powerful current delivery system you need a power amplifier or the adddition of a transistor.