Can the tire type and profile significantly affect balance? I have been riding the same road bike, a Fuji Espree, for a few years and am very used to it: I used to be able to ride it without hands on the handle bar very easily, take 90º turns in city streets because it is well balanced and responds in a predictable way.

I recently changed the old Strada-K Forte tires to new Continental Ride Tour Extra which have a more square profile, deeper grooves.

Since then the balance of the bike has felt off: As soon as I remove my hands from the handle bar, the bike starts steering to the right and even with my hands on it, turning feels harder.
 I'd like to find new tires that won't have that same effect but am not sure what feature is  causing this? My intuition is that the more square profile is causing, do you think that is the case?
I did the exact same thing a long time ago. I wonder what width tyres you used?

Basically, I had a touring bike which I used for "rough" touring, and used the 38c version of this tyre. 

It was fine at lower pressures in rough roads. 

I then bought a set of 25c versions for a winter road bike. Horrible! These, tyres are not designed for smooth roads and high pressures. If someone doesn't understand this problem, basically, the tyre has no smooth corning edge. None. 

Just replace the tyre for something more friendly.

We can recommend tyres for specific scenarios, but you haven't mentioned the type of riding or your preferance in setup.