Too Much of the Whites in my Eyes Show How Can This Be Fixed?

My initial reaction would be to leave your eyes as they are.  However, if you want a second opinion, I sometimes refer patients with your concern to a colleague across the street from my office in New York City. 

His name is Dr. Glen Jelks, and he may have had experience with your problem before.

Dear Shonrea,
A full face photograph would be very helpful as how you position your face will have an effect on the degree of white showing under the colored portion of the eye. This show of white is called inferior scleral show.
Normally, the lower eyelid margin rides the eye just at the inferior limbus. The limbus in the circle defined on the eye where the colored portion of the eye meets the white portion of the eye. So yes, it is true that your lower eyelid rides very low on the eye. Another way of thinking about this issue is that the eye is very prominent relative to the projection of your cheek bone.
Now to understand the issue a little more fully, it is important to understand a little bit about the geometry of curved objects like the surface of the eye. When a plane flies form say Los Angeles to London, the pilot does not fly a straight line from one city to the other. This is because on a spherical object like our planet earth, the shortest distance between to points is not a straight line but rather a curved line, called a geodesic. So the pilot flies what is called a polar route between Los Angeles and London. This curved path follows a geodesic arch between these two cities which defines the shortest distance between these cities.
The same phenomena is important to understand when it comes to eyelid surgery because the eye is not flat but instead is spherical. You may consult plastic surgeons who offer to perform a procedure called a canthopexy or canthoplasty. These are slightly different procedures that involve tightening or shortening the eyelid with the intent of lifting the lid margin higher up on the eye. Well this will not only not work, but this will actually force the eyelid to ride even lower on the eye due to the spherical shape of the eye. It is kind of like what happens when someone tightens their belt and the pants and belt slide under the fullness of the belly. Ouch.
As a result of this under appreciated geometry, as you can image, a lot of well meaning plastic surgeons run into a lot of trouble with this type of lower eyelid surgery. I end up fixing these types of problems. In your case, you have not had surgery, this is how you come and it is the relative position of your eye to the edge of the cheek bone that accounts for this situation.
It is helpful for someone like yourself to explore how much lower eyelid elevation you will tolerate. Recognize that big eyes are considered to be beautiful. When the eyelid is lifted up, the eyes will be perceived to be smaller. It is important as part of the consultation to explore the happy balance between the apparent bigness of the eyes and the desire to have the eyelid cover more of the eye. In terms of how to actual raise the lower eyelid, it is important to elevate the cheek.
In your case with means improving the projection of the cheek bone at the orbital rim. I do this with a swinging eyelid approach working primarily behind the lower eyelid so there is virtually no skin incision. A custom hand carved ePTFE orbital rim implant is made at the time of surgery and this extends the projection of the orbital rim. Next the cheek is lifted onto this implant. The lower eyelid needs to be lengthened across the entire eyelid. This is done by using hard palate graft from the roof of the mouth. Finally the canthal angle is reformed and the eye is sewn closed for 1 week. The other side is done at a separate surgery one week later. Obviously this is a very big undertaking.