All major infrastructure such as roads, power, communications etc needs to be owned and run by government upon which is then a level playing field on which free enterprise can then compete.
When you have any major infrastructure essential to running a business in private hands you no longer have a democracy or free enterprise.
Main difference between digital platforms and government owned utilities it that utilities require access to land. No one is stopping you from rolling out your own digital platform, while there is no practical way to get any land to lay your own gas pipeline. Therefore the answer to your question is "no, Google search cannot be considered public infrastructure, and one for the reasons is that it does not require any public land to operate".
When you have any major infrastructure essential to running a business in private hands you no longer have a democracy or free enterprise.