Are there any microcontrollers with integrated H bridge for DIY servo?
[PLEASE NOTE:  The question was altered AFTER I answered!   Check the original post before you get cranky and down-vote!  Sheesh!]

The motor driver you selected doesn’t have any “brains,” it is merely a device to control the speed and direction of the motor depending on the input control signals.

A (very) simple servo control system diagram:



The project you’re developing is one of those you work bassackwards on.   You know the load to drive, then you must select the gearbox, motor and position feedback device.  (A potentiometer might be fine for very light use, say 100 operations a day, maybe 20 days a year, but will wear down to the nub rapidly if used 100’s times a day for months on end!   You’ll be looking at a magnetic rotational position sensor or optical position sensors (http://en.wikipedia.org/wiki/Rotary_encoder)  – something that will not wear out in a couple of days!)   

Next comes the “brain.”  A microcontroller (or equivalent device, the "MCU" in the datasheet you provided) that will compare the input [control] signal (where you desire the output to land) to the actual position of the output shaft.   The “brain” calculates the difference between the input and actual position then orders the motor controller (h-bridge/amplifier) which way to turn (and in some instances, the speed to get to that position)  known as PID (http://en.wikipedia.org/wiki/PID_controller).

[I do not know of a single integrated circuit package (IC) containing both a MCU and h-bridge.  SEE EDIT BELOW FOR LINKS:]

EDIT:  http://www.jrkerr.com/index.html  http://www.jrkerr.com/pssc_bd.pdf