Yes it is possible. The drawback is that the modal responses are coupled by the damping matrix, so you'll need to use the formulation of Newmark's method that uses the full system M, K, and C matrices - i.e. if you have a 3-DOF system you need to do one matrix calculation rather then summing 3 scalar calculations.

The matrix formulation isn't too much different from the SDOF formulation, but if you're using Excel it may be a bit painful. If you have access to Matlab or MathCAD, or a free alternative, they should be able to make short work of it.

My go-to text for this is Chopra, third edition, which covers this in Section 15.2. You can also probably find a good online reference fairly easily.