The body does, in fact, take calcium from the bone to balance the blood pH.
When the calcium is low in the blood the parathyroid gland secretes Parathyroid Hormone to take calcium out of the bone and bring it into the blood. It also tells the gut to absorb more calcium and Vitamin D, and the kidneys to absorb more calcium.
This is basic physiology. Bicarbonate ions also factor into the picture and that is the part that you presented.