@Bill_Thayer No, I'm sorry, but this information cannot be extracted. One thing to know, however, is that the semivariogram is estimated using no more than 1000 input features. If the input features have more than 1000 valid records, a random sample of 1000 is taken, and only these features are binned for semivariogram estimation*. So even if you could extract the actual number of binned pairs, it would not match the number you expect, unless you have fewer than 1000 points.
If this information is something that would help you in your work or research, your best bet is submit the idea through ArcGIS Ideas.
*After estimating the semivariogram from the random sample, all input points are used when making predictions, not just the 1000 point sample.
