No, none at all -- and if you had asked the right question the first time out ("can I use a Google custom search widget here?") then nobody would have raised any objections, technical or otherwise. All you need to worry about is Google's own terms of service (which means that it has to be obvious that you are using Google search and not claiming their results as your own).
When you asked if you could host an actual search engine, people assumed that you wanted to build and host a search engine. That involves creating bots to crawl the web (which, except for very narrow searches, such as I described in a previous post, would break the "continuous script" clause of the Terms of Service and Acceptable Use Policy), indexing all of the information in a database (which would have put you over the High Resource Usage limits constantly), parsing queries and retrieving relevant results. A lot of people think they can do stuff like that easily, but they underestimate the requirements and overestimate their ability. Spider bots and algorithms like MapReduce and PageRank are only part of the problem.
Sergey and Larry were lucky as well as brilliant -- the web was a lot smaller and expectations a lot lower when they first created Google. It would be hard for just a couple of guys to do the same thing today. Then again, the web wouldn't be what it is today if they hadn't made finding things a lot easier. (God, I remember Alta Vista and the original Yahoo! Not fondly, either.)