Why would diet Mountain Dew make my blood sugar high?

It doesn't. Diet mountain dew does not contain many calories at all, and no carbohydrates. Therefore, it cannot raise your blood sugar. I would look into other foods you may have eaten around the same time for a culprit as to why your sugar levels may be rising.
It doesn't. I would look for other reasons why your blood sugar is high, such as 'hidden' carbohydrates in fruits, juices, and starches.
Insulin Resistance. You probably have some baseline insulin resistance. Perhaps it was some other factor of your meal or overall lifestyle and exercise level. You did not mention a diagnosis of diabetes, or what your weight is but it sounds like you are watching your diet and nutrition. I am glad you are engaging in your health and accepting responsibility for it!