I doubt that. I think it will split the EU even further. The UK has already left, and Italy is seriously pissed off at the lack of help from the EU.
There is already a fund/organisation that is designed to help move supplies, money, resources, whatever is needed in times of crisis. A country in need requests help, and other countries send what is needed. How many countries do you think helped Italy when they requested it in the early days of the outbreak? None. They ended up getting supplies from China because no other European country would give so much as a face mask. I'm pretty sure that won't be forgotten.