Fallback disks are expected to form around young neutron stars. The presence of these disks can be revealed by their blackbody spectrum in the infrared, optical, and UV bands. We present a re-reduction of the archival optical and infrared data of the Vela pulsar, together with the existing infrared and UV spectrum of Vela, and model their unpulsed components with the blackbody spectrum of a supernova debris disk. We invoke the quiescent disk solution of Sunyaev and Shakura for the description of the disk in the propeller stage and find the inner radius of the disk to be inside the light cylinder radius. We perform a high-resolution X-ray analysis with XMM-Newton and find a narrow absorption feature at 0.57 keV that can be interpreted as the K-alpha line of He-like oxygen (O VII). The strength of the line indicates an element over-abundance in our line of sight exceeding the amounts that would be expected from interstellar medium. The spectral feature may originate from the pulsar wind nebula and may be partly caused by the reprocessed X-ray radiation by the fallback disk. We discuss the lower-than-three braking index of Vela as partially due to the contribution of the propeller torques. Our results suggest that the pulsar mechanism can work simultaneously with the propeller processes and that the debris disks can survive the radiation pressure for at least similar to 10(4) yr. As Vela is a relatively close object, and a prototypical pulsar, the presence of a disk, if confirmed, may indicate the ubiquity of debris disks around young neutron stars.