I think it would be a good idea to require a warrant in such cases. But it’s not a 4th amendment issue.
The obvious issue with this is that (for example) it seems to treat a Google Docs spreadsheet differently than one stored locally. The way the law is currently worded simply isn't sufficient (IMO) now that the internet exists.
Yes, there is a bunch of fancy technology that makes it look and feel like the two things are the same. But why should a technological illusion trump physical reality? In reality, you can tell where the bits are and who owns the physical thing that the bits are on. The fourth amendment very neatly and cleanly delineates between the two scenarios based on who owns the relevant “papers and effects.”
Reflecting a bit more on my Google Docs example (which differs from the original browsing history topic) I believe it was faulty. It would be a bit odd to treat bits I own as not mine simply because they're stored on hardware owned by someone else. In fact on it's face it appears to go against existing precedent that the 4th amendment applies to a rented storage unit.
When it comes to sensitive data like browsing history though, regardless of legal ownership it certainly feels like a form of unreasonable search. I really can't blame the original authors for failing to anticipate the existence of the internet though.