Kane was selected by Mauricio Pochettino to face Liverpool in Saturday's showpiece despite having been out since early April with an ankle injury.
Mohamed Salah converted a contentious penalty in the second minute and Divock Origi struck late after coming off the bench to earn the Reds a 2-0 win and their sixth European Cup.
Kane struggled to make an impact, contributing just 11 passes and 26 touches and only managing to record a single shot on goal despite playing the full 90 minutes.
But as Kane thanked Spurs supporters for their backing, the England captain promised his team will bounce back from losing their first ever Champions League final.
"Gutted we couldn't get the job done last night. We'll learn from it and come back stronger next year," Kane wrote on Twitter.
"Thanks to our fans for your unbelievable support home and away this season. You've been incredible."
Kane will hope to put the disappointment of Madrid behind him when he leads England into the Nations League finals.
The Three Lions, bidding for a first major international trophy since winning the World Cup in 1966, face Netherlands in the semis.
Kane will not be joined by team-mate Kieran Trippier, though, after the right-back was left out of Southgate's squad in favour of Kyle Walker and Trent Alexander-Arnold.
Trippier admitted on Saturday he deserved the snub following a number of poor performances for Spurs.
The Manchester City academy graduate has been among a number of players to suffer with injuries this season.
Kane himself had not played since the April 9 defeat to eventual Premier League champions City before Saturday's final.
And many of Tottenham's players appeared weary in Madrid as Spurs put in a lacklustre performance on their way to runners-up spot.
Pochettino insisted he was still proud of his squad, who managed a stunning late comeback against Ajax in the semi-final to take Spurs fans to Madrid.