How do I provide constant voltage to a load?
Your voltage divider has equal resistances above and below its midpoint. Since the current is the same through both, the voltages are the same across each, and you find yourself dividing the voltage in half: 2.5V and 2.5V.

Now, if you add a current drain from the midpoint to ground, you change the currents and thus the voltages. And yes: this breaks your 2.5V "supply".

If your microcontroller indeed has a constant input impedance of 200Ω (drawing a constant 12.5mA at 2.5V), then you could just supply it from the 5V supply through a 200Ω resistor. Bingo: you've got a new equal divider, and the microcontroller will get its 2.5V. But I seriously doubt that a micro would draw a constant current; the current almost certainly depends on what its code is, and if you're driving any external loads then all bets are off.

Another option would be to return to your original voltage divider, but make sure that the current through the dividers is much larger than the current through the microcontroller. Example: if you used two 10Ω resistors, then adding the 200Ω micro load would only pull the center voltage down from 2.5V to 2.44V, and variances in the micro's load wouldn't change that much. Then again, you'd be drawing 250mA through the divider, and you'd have to use 1W resistors. Probably not a great tradeoff.

So, I'd follow @Ignacio Vazquez-Abrams' advice, and get yourself a 2.5V regulator.