Unless you're using Apple's base libraries. Implementing a new Base Library is hard, I've gotten a tiny subset working on my own to see just what is involved. I wouldn't recommend writing the full thing unless you have a burning desire to do it. Porting GNUstep Base Library to various platforms is possible, and has already been ported pretty widely. You'll have to bundle LGPL libraries with your application if you plan to distribute binaries, depending on your application that can make Obj-C a non-starter.
I use C everywhere, and never C++. C is not the easiest thing to do, but putting together portable frameworks for C is less work than doing the same for Objective-C. (at least in my personal experience, your mileage may vary). C isn't an object oriented language, in that it doesn't do anything special to make that work. But there are OO toolkits for C, with varying degrees of horribleness. A classic is Xt (X Toolkit) and Xaw(X Athena Widgets), which is not terribly C99 friendly anymore. But it does permit limited object-oriented hierarchies. GObject (GTK+) is another option for C, I'm not a fan of it, but it's not obsolete unlike the Xtk/Athena example I mentioned.
What confuses me about your question is why are you only asking about C++ or Objective-C? There is Free Pascal (Delphi clone) and D. (D is very good, I highly recommend that over Obj-C). There are probably a lot of other options out there that would suit your initial criteria, especially surprising is you didn't mention Java. It's very good example of OO, perhaps OO taken to the extreme, but it's fairly clean. (Obj-C is a little hacky, but thats it's charm and power)