Isn't the 22nd amendment about limiting the number of times a person can be elected to the office of president, to twice?

It doesn't limit the office of vice president, so could the following scenario happen?

Obama is picked for vice president, and after the the inauguration, the president is out of the picture for some unforeseen reason.

Could Obama, then, be elevated to president, but not elected president, and thus no violating the 22nd amendment?

Here is the the text of the 22nd amendment:

"No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once."