Drawing on statistics available at Baseball-Reference.com, I have plotted complete games as a percentage of games started, by league and for both major leagues, at five-year intervals from 1904 through 2009. (It would have been too cumbersome and not worth the effort to have transcribed the statistics for every season from 1901 through 2010).
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.