The word gate is a good old English word, the form being geat in Old English (plural geatu) and whose meaning was ‘an opening or entrance’. A watergate (from the 15th century) is a channel for water; the first element of the compound, water, is also a native English word.
Its use as a suffix is fairly recent, dating to the infamous Watergate incident of the Nixon administration. The shorthand referent for this major political scandal became the name of the hotel itself — the location of the crime’s execution. Pulling the compound apart, the second term (-gate) has become a highly productive ending, now carrying the meaning of ‘scandal’. Amazing little cognitive move, isn’t it? Wikipedia lists dozens of coined terms for scandals that now use this term, including a few cases of fictional scandals from sitcom episodes.
Here are just a few among the more famous recent scandals:
- Spygate
- Climategate
- Irangate
- Tigergate
- Tasergate/Troopergate
- Angelgate
Note how the semantics of ‘political scandal’ has generalized to ‘personal scandal’ (Tigergate) and ‘financial scandal’ (Angelgate). The form’s usage is robust enough to include variants for a single incident (Tasergate/Troopergate). It will be interesting to see whether the suffix’s course through English becomes even broader in time to encompass more general incidents that don’t convey the notion of ‘scandal’. Will the suffix -gate expand to include affairs and events that don’t convey a pejorative sense, perhaps including some with a positive meaning?
Word Travels 
Prior to Watergate, the most sensational political scandal in American history was Teapot Dome, a half-century earlier during the Harding administration. Yet ‘dome’ didn’t become a suffix denoting scandal…
It sounds silly, but I wonder if the fact that the letter group ‘gate’ already constitutes the ending of a number of English words—vulgate, aggregate, congregate, conjugate, derogate, abrogate, profligate, negate, legate—might subconsciously have facilitated its mobilization in the way you describe…
To be fair, I should say that broadcasting was barely into its infancy at the time of Teapot Dome, and so there was not the mechanism for amplification there was with Watergate. No American who was an adult at that time will soon forget the televised Senate hearings held by Sam Ervin’s committee in the summer of 1973; the extraordinary disclosure (by Alexander Butterfield) that Nixon had maintained a taping system in the White House; John Dean’s warning to Nixon that there was a cancer growing on the Presidency; the Saturday Night Massacre, in which the Attorney-General of the United States and his deputy resigned rather than fire the special prosecutor; and at the last, inevitably, that bright morning in August 1974 when Richard Nixon flew out of Washington for the last time, the phenomenological end of what we now call ‘the Sixties,’ which had begun a decade earlier after the Kennedy assassination…
So, together with the increased malleability of American speech (also mediated by broadcasting), that’s a pretty powerful launch for the suffix -gate…
Well put!