Great question! Because apparently "the web" doesn't have the answer. But to save you the trouble, I wanted to share what I DID find:
Top 10 Songs of 1968 featured on the Billboard Top Pop Hits CD:
1. Classical Gas - Mason Williams
2. A Beautiful Morning - The Rascals
3. Honey - Bobby Goldsboro
4. Love Is Blue - Paul Mauriat
5. Young Girl - The Union Gap
6. Spooky - Classics lV
7. Harper Valley P.T.A.
8. Abraham, Martin And John - Dion
9. Slip Away - Clarence Carter
10. Hush - Deep Purple
and on the Top Rock n Roll CD from Billboard:
1. Tighten Up - Archie Bell & The Drells
2. Judy In Disguise (With Glasses) - John Fred & His Playboy Band
3. (Sittin' On) The Dock Of The Bay - Otis Redding 4. People Got To Be Free - The Rascals
5. Chain Of Fools - Aretha Franklin
6. Green Tambourine - The Lemon Pipers
7. Cry Like A Baby - The Box Tops
8. Mony Mony - Tommy James & The Shondells
9. Yummy Yummy Yummy - Ohio Express
10. The Horse - Cliff Nobles & Co.
ONE of those songs was #1 on July 7, 1968.
and on the "This Day in Music History" site (resourced below), there are matches for top singles for July 7th, but not for the year 1968.
and Finally, on July 7, 1968, Led Zeppelin was officially founded by Jimmy Page. And THAT, is worth a mention.
My guess is that the #1 song on Billboard's chart for July 7, 1968 was already on the charts for several days or weeks.
Check out the Rock n Roll Hall of Fame's timeline, below. It's a time-killer, but your answer may be in there.