Some redness will disappear on its own without applying eye drops. Remember that don't rub your eyes with your dirty hand. But some kind of infection need professional treatment and doctors may require you to apply eye drops. if you got the contagious pink eye. You can't refuse to use eye drops. Because eye drops contain anti-viral ingredients that are good for your infected eyes. The artificial can moisten your eyes but they can't kill virus, so they are not as good as special eye drops at this moment.
Many eye drops can remove eye redness, but they are just contract our vessels in our eyes and reduce the symptoms. However the inflammation is still in our eyes. I will not suggest you to use this kind of eye drops. You can use artificial tears to moisten your eyeballs. Or you can put a piece of ice on your eyes. The coolness is good for the inflamed eyes. If your red eye are just caused by lack of sleep, you symptom will totally disappear after you catch up enough sleep.