O(1) doesn't refer to any length of time. A single operation could take .001 ms on a webservice and 100 seconds in a database and they both could be using O(1) functions:
http://en.wikipedia.org/wiki/Big_O_notation
It's hard to know quite what you're asking. If you're asking whether accessing a local database is generally faster than accessing a similar service over the internet, then I expect that, generally, the answer is that the local database will be faster. The call over the internet to the web service has a lot of overhead and communication over internet is relatively slow. Evan on a slow computer a databases can perform many thousands of simple queries per second. Contrast that with access over the internet, where you'd be lucky to get 50 round trip requests per second, not even accounting for time it takes to perform the requested operation on the server.
If you're asking whether a server on the web can serve data faster by avoiding a database and calculating results directly, then the answer is it depends. The call to the database in this case adds unnecessary overhead if the data in it can be easily calculated in a stand-alone function. The answer to this question doesn't really have anything to do with a "web service". Is it faster to calculate an answer in a function or to access the answer using a query on a database? As I said, the answer would depend on the complexity of the particular function you had to use, and weighing its computation time against the overhead of accessing the answer (or part of the answer) directly from a database.
In short, the answer to your question depends on what exactly you're asking. It would also probably help to know why you're asking the question. I have a suspicion that the real answer is that this probably isn't something you need to worry about, not really a practical concern unless you have a particular situation requiring optimization.
If you're concerned about comparison of speed when webservice and database are both on a lan, I'm pretty sure the overhead of the db is a less than the webservice. The application typically maintains a stateful connection(s) to the db, while requests to a webservice are via http, which is stateless, relatively higher overhead, and slower. Could be wrong, though. Best answer would be to whip up a simple webservice, query, and (1) measure time it takes to retrieve results using both methods, and compare, and/or (2) create an app that opens a lot of threads and do some load testing.
A caveat: If your app doesn't maintain an open connection or have access to a pool of connections with the db, then the db alternative may well be slower. Initial creation of a db connection can be relatively slow. But that shouldn't figure into things, since you should write your app so that an open connection is always maintained.