Before you say yes, I'd consider anything non-physical unable to physically interact with the universe.
If God is non-physical and does interact with the universe, how is this possible if God is not physical.
OTOH, if God is physical, shouldn't we be able to detect/measure God?
According to Einstein matter (mass) cannot move at the speed of light. If you plug into the equations for Special Relativity, you get an infinite energy requirement. The speed of light is the cut off for the state we call matter. If God is not matter, he would need to exist on the other side of that discontinuity and have a reference at the speed of light. This science is consistent with God being the light of the world.
Energy, such as light, moves at the speed light but is also able to impact matter. Energy is the bridge between God and the material universe. In Genesis, God was brooding over the deep; in his speed of light reference He says let here by light! This made a bridge between the two references, allowing God the spirit, to interact with the material creation.
In inertial references or the reference of matter, we live in space-time, where space and time are integrated. At the speed of light reference of God, inertial space-time, breaks down into separated time and space. One can move in time, without the constraint of space and one can move in space without the constant of time. The former allows one to know the history of the universe in all places at the same time; omniscience. While the latter allows one to be anywhere in zero time; omnipresent.
The interaction of the God reference, with inertial material reference, would amount to disconnected space or disconnected time interacting with space-time. This should give us space-space-time affects or space-time-time affects, respectively. Gravity, for example, is an acceleration, which has the units feet/sec/sec which is one part distance and two parts time; space-time-time. Acceleration is a manifestation of the God reference in inertial and is part of all the forces.
Note: If you plug into the equations for Special Relativity as velocity V increase toward the speed of light space-time or space and time appears to contract. At the speed of light, this contracts our perception of the universe to what appears to be a universe that is only a point-instant in space-time. This reference, at the speed of light, is how omnipresence and omniscience work. If the universe looked like a point, then everything is overlapping and one is everywhere at the same time.