It's not a question of are your caucasion genes being stronger than his mexican genes. It's a lot more complicate that the classic blue eyed vs brown eye baby examples they give in highschool biology.
I'll TRY to explain it with this over simplification (which wil get the point across, but may not be scientifically accurate).
The DNA in every cell of our body consists an alphabet of 4 letters: A,C,D, and G. These letters are stacked up in pairs, with A always paired with D and C always paired with G. So a stran of DNA might look like this:
AD
CG
GC
DA
Reading down the sequence of pairs is what provides the instructions for what we look like. For example, someone with the above DNA sequence "AD,CG,GC,DA" might have blue eyes, while someone with the DNA sequence "GC,GC,AD,CG" might have green eyes.
We grow new cells by existing cells dividing in half. Each half gets either the string of letters on the left of the string of letters on the right, so for the DNA of "AD,CG,GC,DA", one side woudl get the sequence ACGD while the other half would get DGCA. The two new halfs would grow, and as they did so, they would add the missing pair to it's DNA sequence. So each A woud grow a D beside it, each C a G, each D an A, and each G a C. Because of this mirror effect, when the two cells have finished growing, they will have identical copies of the original DNA.
Now to make the next part easier to understand, lets give each pair a new name. Lets call AD=1, CG=2, GC=3, and DA=4. So another way we could represent the sample DNA would be "1234". That means our green eyed sample would have the DNA "3312"
Sex cells are special. When they are created, the DNA doesn't split down the middle, it splits across the middle and never grows back. So a woman with DNA of "1234" will produce two types of eggs: "12" and "34". If her husband is the green eyed sample, he will have two types of sperm: "33" and "12". This means that they can have four differnet types of babies: "1233", "1212", "3433", and "3412". What can happen is that "1233" will be blue eyed, "1212" will be brown eyed, "3433" will be green eyed, and "3412" will be green eyes.
So did you catch that? A blue eyed wife wit a green eyed husband can produce green, blue, or brown eyed babies.
So it is with you. This baby might be white skinned, but the next one might be brown skinned.
Now while I'm correct is saying that our DNA alphabet has only 4 letters, I've simplified the lenght. I've given examples where the DNA consist of only 4 pairs of letters strung together. Our actually DNA consists of billions of pairs strung together.
And as complicated as I know I've already made this, it gets even more complicated from there. Scientist have started to discover how the same DNA can make both a brain cell, a skin cell, and a muscle cell. It has something to do with chemicals that wrap around the DNA. And because these wrapping chemicals can be influenced by our environment, even identical twins won't be perfectly identical. They will both have the same DNA, but because even conditions on one side of the womb will not be exactly like conditions on the other side, there will be differences. This is why it's possible for two people to be identical twins, yet one be diagnosed with sever autisum, while the other is perfectly normal.