The result:

Observations:
The rise of complete games in the American League following the introduction of the designated hitter was a transitory phenomenon.
The statistics for the National League are therefore more indicative of long-term trends.
The complete game has been on the wane since the early 1900s, but the trend has accelerated since 1970. (The use of the logarithmic scale for the vertical axis highlights that acceleration.)
It is likely that the incidence of complete games has reached a minimum, at around three percent of games started. That is to say, the percentage is unlikely to drop further because there will always be those (infrequent) occasions on which a starter is still throwing well and has thrown fewer than 100 pitches as he goes into the eighth and ninth innings of a game in which his team is leading, tied, or only a run or two behind.