Does passive perception supersede active perception?

If a player makes an active perception check and it is lower than their passive perception, should we use the passive perception instead?

During our weekly game yesterday the party scout was actively searching for hidden monsters. He rolled poorly and ended up with a 7 on his perception check, seeing nothing. The claim was then made that since his Passive Perception was a 15, he would intuitively notice anything that he didn’t actively spot.