How repeatable is the output from an opamp
The main specifications you are looking for are 'long term stability' and 'temperature coefficient of offset voltage'.

Where you are looking to make a very precise comparison against a reference, it is common to use a precision amplifier with a modest gain, like 100x, ahead of a more conventional comparator. 

For example, look at the OP177 with its max specification of 0.3uV/C for input offset versus temperature, and a typical input offset drift of 0.3uV per month. Those are an order of magnitude or more better than cheap 'cooking grade' opamps and comparators. This is my 'go-to' device for low cost (digikey $6 in ones for highest F grade) high precision work. If an amplifier or comparator doesn't specify drift/time, then you can assume it's worse than that!

Indeed given a stability of the order of uV in 2.5v, your difficulty is not going to be the comparison system, but the reference voltage. The trick to getting good reference stability is to a) choose a device well specified for drift (buried zener is the type to look out for) b) run it continuously 24/7/365 c) from a regulated voltage d) at a constant temperature. Users often use two cascaded stages of voltage and temperature regulation, as it eases the stability requirements for those.

How do you check that you have a stable voltage? Build two or three systems, and compare them. 

Mind you, one you have a decent low drift amplifier, and a low drift reference, is the stability of the process that you're measuring meaningful at this level?