You cannot uninstall GRUB. As your installation stands, GRUB is necessary to boot Ubuntu (that's why it's called the bootloader). Every OS has a bootloader, and every OS needs that bootloader to boot (lots of booting :P).
As people in the comments have pointed out, there are alternatives to using GRUB. However, there is no reason to switch to one, since you just want the GRUB screen to be hidden. I believe that all of the other bootloader options work in similar ways, so you shouldn't worry about installing something else.
I believe what you want to do is hide GRUB. This is possible, and quite easy. Press Alt+F2 and paste this before hitting Enter:
gksudo gedit /etc/default/grub.
A text-editor window will open, showing something like:
GRUB_DISTRIBUTOR=”`lsb_release -i -s 2> /dev/null || echo Debian`”
Remove the two
GRUB_HIDDEN_TIMEOUT_QUIET. Save the file, open a terminal and run
sudo update-grub. Reboot and you should have no GRUB window show up.
1 to avoid potential problems. GRUB will flash by, but it won't sit there for 10 seconds.
If you want, you can do this graphically.
Run these commands in a terminal window:
sudo add-apt-repository ppa:danielrichter2007/grub-customizer
sudo apt-get update
sudo apt-get install grub-customizer
Once that's done, search for and open GRUB Customizer.
Go to the General tab and deselect
show menu and
look for other operating systems. Click the refresh button (blue circular arrow near the top right) and then click save.
Reboot and see no bootloader.