No. Rabbits belong to the mammalian order Lagomorpha (family Leporidae), not the rodent order Rodentia. They were once grouped with rodents, but taxonomists moved them into Lagomorpha in the early 20th century.
A clear anatomical difference is that lagomorphs have two pairs of upper incisors (a large front pair plus a smaller pair behind), while rodents have a single pair of upper incisors. Rabbits and rodents are still close relatives in the broader clade Glires, but they are not the same order or family.
Source: Wikipedia — Rabbit (taxonomy section; cross-checked with NCBI Taxonomy for Lagomorpha).