Emmanuel Boateng's hat-trick and Enis Bardhi's brace put lowly Levante in charge, and despite Philippe Coutinho completing a treble of his own and Luis Suarez scoring a penalty, Barca's bid for history came up short.
But they fell just short in a match that nearly defied logic - they had only conceded 24 goals in their opening 36 games.
Levante should have scored their sixth goal through substitute, Ruben Rochina, but the Barcelona youth product shot wide after pouncing on a blind pass back to the goalkeeper.
In what would surely be a routine comeback and even win for Barcelona at this point, Levante came flying out for the second-half.
Suarez played in Coutinho for an early chance but the Brazil midfielder overplayed his hand - wastefulness Barca would soon regret.
Luis Suarez has scored in each of his three league appearances versus Levante (three goals).
"We have already talked about him in recent weeks", said Lopez. "The explanation is the same as that of our play over the last 2 1/2 months: work, trust in one another and the belief that this team has no limits".
Barcelona sewed up the La Liga title long ago, but the Catalan giants were chasing an invincible league campaign heading into the penultimate weekend of the season.
Valverde will be questioned - and not only for leaving Messi off his squad.
With Sergi Roberto suspended, Nelson Semedo will start at right-back for Barcelona at Levante, while Thomas Vermaelen looks set to feature with Umtiti carrying a minor injury.
Bardhi restores two-goal lead on 46′.
Barca did manage to pull a goal back five minutes before the break as Coutinho found space on the edge of the box and curled the ball home, 1-2.
Get the latest World Cup news updates. The team won La Liga and the Copa del Rey, and going undefeated would have just been icing on the cake.
Elsewhere, Espanyol thrashed relegated Malaga 4-1 to delight their fans in their final home game of the campaign.