Both Windows (Win32 API) and OS X (Cocoa) have their own APIs to handle windows, events and other OS stuff. I have never really got a clear answer as to what Linux’s equivalent is?
I have heard some people say GTK+, but GTK+ being cross platform. How can it be native?
In Linux the graphical user interface is not a part of the operating system. The graphical user interface found on most Linux desktops is provided by software called the X Window System, which defines a device independent way of dealing with screens, keyboards and pointer devices.
X Window defines a network protocol for communication, and any program that knows how to "speak" this protocol can use it. There is a C library called Xlib that makes it easier to use this protocol, so Xlib is kind of the native GUI API. Xlib is not the only way to access an X Window server; there is also XCB.
Toolkit libraries such as GTK+ (used by GNOME) and Qt (used by KDE), built on top of Xlib, are used because they are easier to program with. For example they give you a consistent look and feel across applications, make it easier to use drag-and-drop, provide components standard to a modern desktop environment, and so on.
How X draws on the screen internally depends on the implementation. X.org has a device independent part and a device dependent part. The former manages screen resources such as windows, while the latter communicates with the graphics card driver, usually a kernel module. The communication may happen over direct memory access or through system calls to the kernel. The driver translates the commands into a form that the hardware on the card understands.
As of 2013, a new window system called Wayland is starting to become usable, and many distributions have said they will at some point migrate to it, though there is still no clear schedule. This system is based on OpenGL/ES API, which means that in the future OpenGL will be the "native GUI API" in Linux. Work is being done to port GTK+ and QT to Wayland, so that current popular applications and desktop systems would need minimal changes. The applications that cannot be ported will be supported through an X11 server, much like OS X supports X11 apps through Xquartz. The GTK+ port is expected to be finished within a year, while Qt 5 already has complete Wayland support.
To further complicate matters, Ubuntu has announced they are developing a new system called Mir because of problems they perceive with Wayland. This window system is also based on the OpenGL/ES API.
Linux is a kernel, not a full operating system. There are different windowing systems and gui's that run on top of Linux to provide windowing. Typically X11 is the windowing system used by Linux distros.
Wayland is also worth mentioning as it is mostly referred as a "future X11 killer".
Also note that Android and some other mobile operating systems don't include X11 although they have a Linux kernel, so in that sense X11 is not native to all Linux systems.
Being cross-platform has nothing to do with being native. Cocoa has also been ported to other platforms via GNUStep but it is still native to OS X / macOS.
Strictly speaking, the API of Linux consists of its system calls. These are all of the kernel functions that can be called by a user-mode (non-kernel) program. This is a very low-level interface that allows programs to do things like open and read files. See http://en.wikipedia.org/wiki/System_call for a general introduction.
A real Linux system will also have an entire "stack" of other software running on it, in order to provide a graphical user interface and other features. Each element of this stack will offer its own API.
To aid in what has already been mentioned there is a very good overview of the Linux graphics stack at this blog: http://blog.mecheye.net/2012/06/the-linux-graphics-stack/
This explains X11/Wayland etc and how it all fits together. In addition to what has already been mentioned I think it's worth adding a bit about the following API's you can use for graphics in Linux:
Mesa - "Mesa is many things, but one of the major things it provides that it is most famous for is its OpenGL implementation. It is an open-source implementation of the OpenGL API."
Cairo - "cairo is a drawing library used either by applications like Firefox directly, or through libraries like GTK+, to draw vector shapes."
DRM (Direct Rendering Manager) - I understand this the least but its basically the kernel drivers that let you write graphics directly to framebuffer without going through X
I suppose the question is more like "What is linux's native GUI API".
In most cases X (aka X11) will be used for that: http://en.wikipedia.org/wiki/X_Window_System.
You can find the API documentation here
The linux kernel graphical operations are in /include/linux/fb.h as struct fb_ops. Eventually this is what add-ons like X11, Wayland, or DRM appear to reference. As these operations are only for video cards, not vector or raster hardcopy or tty oriented terminal devices, their usefulness as a GUI is limited; it's just not entirely true you need those add-ons to get graphical output if you don't mind using some assembler to bypass syscall as necessary.
