You asked

Can I treat an eye injury myself?

Babies and young children like to explore and no matter how careful you are chances are they will injure themselves somehow. If your child’s eye is injured, you should always seek immediate, professional medical attention. Here are some tips on what to do while you wait for medical assistance.
Trauma to the eye:  If your child is hit in the eye, rest a protective shield (a paper cup will suffice) on the bone around the eye, making sure there is no pressure on the eye itself. Get immediate professional, medical attention.
Foreign body: If a foreign body has entered the eye, do not try and remove it as you may further damage the eye or force the object in deeper. Rest a protective shield, such as a paper cup, making sure there is no pressure on the eye itself. Get immediate, medical attention.
Black eye: If your child gets hit in the eye area, place an ice pack or cold cloth over the eye. Get immediate, medical attention.
Chemical burn: If your child has gotten a chemical burn, rinse it with fresh, cold water for at least 20 to 30 minutes. Hold their head under the tap and use a clean container to pour water into their eye, keeping the affected eye open with your fingers. Get immediate, medical attention. 

More questions

Children's eyes are sensitive so it is vital to protect them from the sun's harmful rays.
Your young child can have their eyes tested even if they can't read
Regular eye exams are essential in order to monitor your child's eyesight.
There are some signs to watch out for if you think your child may have a vision problem. 
Should your child get any kind of chemical in her eye you should treat it as an emergency and bring her to the hospital immediately.
You should take your child to their G.P. or to the hospital if you notice any of these symptoms. 
You should seek immediate medical attention if an object seems to be imbedded in your child's eye. 
If your child appears to have something in his eye you should seek immediate medical attention if it is imbedded in the eye. 
As your baby learns to crawl and begins to explore her surroundings you will need to child proof your house to prevent eye injuries.
Babies and young children are always getting up to all sorts of mischief and eye injuries are quite common. It is important to always consult with your G.P. should any eye injuries occur.