What is Linux’s native GUI API?

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.