Real responded thanks to Lucas Vazquez's equaliser before half-time, but failed to break Levante's resistence despite incessant pressure as Bale, Marcelo and Toni Kroos came close in the second half.
And their frustration boiled over when Marcelo was shown a straight red card for kicking out at Jefferson Lerma as the game entered stoppage time.
"We can't be happy with our performance, above all in the first half," said Real boss Zinedine Zidane.
Madrid have now dropped points in their opening two home league games of the season, handing the early initiative in the title race to Barcelona.
Barca could move four points clear of Real with victory at home to Espanyol later on Saturday.
"We don't need to go crazy. We need to have patience and think about the next game, which is just around the corner," added Zidane.
With an unusually early 1300 (1100GMT) kick-off time for Real on the back of an international break, Zidane left many of those who had been away with their national teams in reserve.
Despite Ronaldo's continued absence through suspension, Bale and Isco were left on the bench, whilst Luka Modric and goalkeeper Keylor Navas were left out the squad completely.
However, Zidane defended criticism of his rotation policy.
"I am not going to revolutionise what I want to do with this team due to one bad game," said the Frenchman.
"I believe in this team and we need to play better, nothing more than that."
The European champions were caught cold by a sluggish start as Levante went in front thanks to some awful Real defending.
A Levante long-throw from the right was allowed to bounce inside Madrid's six-yard box before Ivi spun and fired across Kiko Casilla into the far corner.
The hosts were dealt another blow midway through the first half as Karim Benzema limped off and with no other natural striker on the bench, Bale was forced to play as an auxiliary centre-forward.
Madrid ramped up the tempo as the half wore on, but had to rely on a set-piece to level nine minutes before half-time.
Real Madrid's Gareth Bale controls the ball during a Spanish league match against Levante at the Santiago Bernabeu stadium in Madrid, on September 9, 2017
Sergio Ramos's towering header was parried by Raul Fernandez, but Vazquez bundled home the rebound into an empty net.
Bale then had a huge chance to turn the game around, but was denied by a smart stop from the Levante keeper as he tried to curl the ball home with the outside of his left foot.
The Welshman continued to be the focal point of the Madrid attack after the break, but, even after the introduction of Isco, Madrid missed Ronaldo's cutting edge in the final third.
A frantic finish saw Bale head inches wide from Isco's cross at the back post four minutes from time.
Marcelo then saw a fierce effort from the edge of the box turned behind by Fernandez, but the Brazilian appeared to kick out at Lerma as he hit the ground and was shown a straight red card.
Despite their numerical disadvantage, Madrid still nearly grabbed all three points as Toni Kroos's effort from the edge of the area clipped the outside of the post.
Atletico Madrid failed to take advantage of Real's slip-up as without the suspended Antoine Griezmann they drew a blank in an intense 0-0 draw at Valencia.
Diego Simeone's men were denied by fine saves from Valencia keeper Neto to turn behind deflected efforts from Angel Correa and Nico Gaitan either side of half-time.