Serie A was suspended on March 9 due to the coronavirus pandemic, which has hit Italy hard.
A day earlier, Inter suffered a 2-0 loss to Juventus and on March 12, defender Daniele Rugani tested positive for COVID-19.
Godin said there was a push for football to continue until the situation was no longer under control.
"We were exposed until the last moment. They kept pulling the rope to see if you could continue playing, until the situation was unsustainable," he told ESPN.
"The health system collapsed, there are no intensive care beds to attend so many critically ill people and people who may have another illness, they cannot attend to them, so many professional doctors."
Godin added: "We continued playing for several weeks, we continued training, playing behind closed doors, until the positive of a Juventus player was detected and we and the Juventus players were quarantined.
"There the championship was stopped. Surely in that match there were other players who would already be infected, so they directly quarantined all of us."
Italy has recorded more than 92,400 cases of coronavirus, reporting 10,023 deaths.