Yes and no.
You are quite correct that virtual particles are a mathematical device used for doing perturbative calculations in QFT. So in that sense virtual particles don't contribute to the mass of the nucleons in ice. But although virtual particles are just a computational device they are describing something real i.e. the energy in the quantum field. And that energy does have a mass given by Einstein's famous equation $E = mc^2$.
The article is a little sensationalist, but what is says is basically correct. Even before QCD was understood we already knew that energy contributes to mass due to observations of things like the mass deficit in nuclei. Even leaving aside quantum mechanics, for example in special relativistic collisions we know that mass isn't conserved. See for example Is (rest) mass conserved in special relativity?
In quantum field theory the objects we call particles are the states of the quantum field in the free field limit i.e. when we consider a single completely isolated particle. When we have interactions between two of more particles the states of the field change, and in fact we have little understanding of what these states are. However we can approximate them as combinations of the free field states. So when we talk about there being three quarks in a proton this isn't really true because the field states in a strongly interacting system aren't the same as the quark states we would get in the free field limit$^1$. It is therefore little surprise that the mass of a proton is not simply the mass of the three quarks inside it.
I feel the article you cite gets a little hysterical about the idea that energy has mass. That has been known for a hundred years and won't be news to anyone who works in the area. However I can see that this seems bizarre to many, even given that much of quantum mechanics is already on the bizarre side.
$^1$ a hypothetical free field limit in the case of quarks