This is actually an interesting proposal. In fact, many utilities went the way of nationalization like water and electricity. Searching the internet, socializing and ensuring a fair market are all also things which could in theory be nationalized given they fulfill a basic need.
Of course, as they are, they would grant whichever government they were given untold power over the entire internet and our lives. Which seems rather... unbalanced. Moreover, no government should retain that right given the internet transcends borders. No one owns all of it.
Letting the free market run its course with no breaks clearly didn't work particularly well either.
Perhaps a third option? Instead of one government ruling all of it. Perhaps they were to be owned by a supranational body where several governments can propose and discuss changes/regulation and keep balances on each other? UN style? Worthy of discussion.
If anyone has other ideas I'd love to hear them.
PS: (Also, when one suggests nationalizations such as this, one does not intend for a nationalized framework to be the ONLY one. Alternatives brought upon by the free market would still certainly compete with any such services.)
Relevant XKCD, as per usual: https://what-if.xkcd.com/96/