Ultimately it would be up to the group you're playing in to decide, but per the RAW no, thaumaturgy can't be used for Counterspelling.
Again, according to the RAW, one of the the four types of Evocation effect is Counterspell, the others being Attack, Block and Maneuver.
According to the Counterspell description in the Evocation section, a Counterspell is an Attack against the energies of the spell the practioner is attempting to counter. This suggests to me that a Counterspell is an immediate collection and use of energy, hence Evocation.
Now one of the other things to keep in mind with Counterspelling, is that it isn't the same as Dispelling something. By that I mean that a Counterspell can negate a spell with immediate effect or a spell with a duration, but as soon as the Counterspell is ended, any ongoing ritual magic with a duration will resume.
I could be mistaken, but I'm assuming that what the OP was really wanting to do was to Dispel magic, permanently ending whatever ongoing magical effect occurring, and not Counterspelling said magical effect, which means that as soon as the Counterspell stops, the ongoing effects resume unless they ran out of duration. Taking a Ward as an example, four 'extra' shifts of power would keep a Ward going for a month, and I can't see a practioner managing to keep a Counterspell going that long.
-Cheers