What is the most secure operating system that I can use to store bitcoins without having to fear a hacker stealing them?
The most secure operating system is no operating system. As Pieter mentioned in the comments, dedicated hardware wallets prevent private keys from leaking out of the secure hardware enclave, and therefore can't be grabbed from the Internet.
All operating systems are enormous, complex software systems with lots of ways to fail catastrophically. It is likely that all common operating systems have undiscovered security flaws. If you have to store bitcoins on a server, I would suggest Linux, since most malware targets Windows. But still, if planning on storing any amount of bitcoins large enough to worry about losing them, don't get your computer near the Internet.
The most secure OS for handling bitcoins (as opposed to “the secure OS” which is an impossibility) is by far Qubes OS. To be precise you can use a split Bitcoin Wallet in Qubes OS, to protect your bitcoins by having your wallet split into an offline “cold storage” wallet and an online “watching only” wallet. Because of the way Qubes OS works, if some malware gets introduced into one of your virtual machines, it wont have the ability to go to the offline virtual machine specifically dedicated to the offline cold storage, thereby protecting your precious Bitcoins. The documentation for setting up a Split Bitcoin in Qubes can be found here.