The incident happened at the Copps Coliseum in Hamilton, Ont., QMI Agency reported.
Eric Michael Costello, 50, and his 7-year-old daughter Janey were standing near the stage holding a sign that read "Smash your guitar, Pete."
Townshend then seemed irritated and then mouthed the foul language at the father and daughter. Videos of the incident were then uploaded to YouTube.
"Apologies and condolences from my side, and very best wishes to you and your family," Townshend wrote in a personal letter to Costello Thursday, asking if they could "meet and be friends and put this behind us."
Townshend said he reached out to a friend to help track down Costello.
"I asked my friend Mike Tyler to find you so I could tell you that I am so sorry the story about us blew up the way it did, but also to tell you that I sympathize with you both, and your daughter's mother, and that I am most certainly not angry," Townshend wrote. "I was not angry on stage either. I was in rock star mode, the big mouth."
Tyler is a DJ for the classic rock station Giant FM 91.7 in Hamilton who met Townshend by chance in Antigua.
"I was surprised at the whole thing because it's really nothing like Pete," Tyler said of what happened at the Hamilton concert. "He is a very friendly person. He invited my family to the Hamilton show and genuinely felt bad about what transpired and wanted to make it right."
Townshend ended his letter by making a promise: "I will not swear."