Is Micro-processor completely Digital?
This is a complex question, because what actually makes a part "digital" can have multiple definitions.

Fundamentally, reality is analog (at least at the scales which most microprocessors operate at). Therefore, you can make a coherent argument that there are not actually any digital microprocessors. "Digital" is a theoretical mechanism for simplifying the expression of analog systems where the analog voltages therein are (as much as possible) constrained to two states, each of which represent a boolean value.

This simplification makes it much easier for our puny human brains to contemplate complex systems, and much easier for people to write software to evaluate the behaviour of said complex systems.

However, if you are asking if any components inside most microprocessors operate outside this simplified view, the answer is generally no. 


Some microprocessors have integrated ADCs (analog-digital converters), which by definition must operate at least partially outside of the digital simplification.
Some microprocessors also have DACs (Digital-analog converters), which are much the same as ADCs.
Some microprocessors have analog comparators that can be configured to act upon input analog signals.
Schmitt trigger input buffers are also partially analog.


Basically, at this point, the question is more, assuming you're asking about whether components inside a MCU operate outside of the digital simplification, the question then becomes "How do you define a microprocessor"? Fundamentally, the *CPU core( of almost all microprocessors is purely digital. 

However, many, many microprocessors integrate on-die peripherals like the ones mentioned above that are very much "analog" devices, so you must ask if you are defining the entire integrated-circuit as the "microprocessor", or just the actual processing core, which may only be a small part of the actual processor's IC die.