MicroRNAs (miRs) are a group of small, 17–25 nucleotide, non-coding RNA that regulate gene expression at the post-transcriptional level. To date, little is known about the molecular signatures of regulatory interactions between miRs and apoptosis and oxidative stress in viral diseases. Lagovirus europaeus is a virus that causes severe disease in rabbits (Oryctolagus cuniculus) called Rabbit Hemorrhagic Disease (RHD) and belongs to the Caliciviridae family, Lagovirus genus. Within Lagovirus europaeus associated with RHD, two genotypes (GI.1 and GI.2) have been distinguished, and the GI.1 genotype includes four variants (GI.1a, GI.1b, GI.1c, and GI.1d). The study aimed to assess the expression of miRs and their target genes involved in apoptosis and oxidative stress, as well as their potential impact on the pathways during Lagovirus europaeus—two genotypes (GI.1 and GI.2) infection of different virulences in four tissues (liver, lung, kidneys, and spleen). The expression of miRs and target genes related to apoptosis and oxidative stress was determined using quantitative real-time PCR (qPCR). In this study, we evaluated the expression of miR-21 (PTEN, PDCD4), miR-16b (Bcl-2, CXCL10), miR-34a (p53, SIRT1), and miRs—related to oxidative stress—miR-122 (Bach1) and miR-132 (Nfr-2). We also examined the biomarkers of both processes (Bax, Bax/Bcl-2 ratio, Caspase-3, PARP) and HO-I as biomarkers of oxidative stress. Our report is the first to present the regulatory effects of miRs on apoptosis and oxidative stress genes in rabbit infection with Lagovirus europaeus—two genotypes (GI.1 and GI.2) in four tissues (liver, lungs, kidneys, and spleen). The regulatory effect of miRs indicates that, on the one hand, miRs can intensify apoptosis (miR-16b, miR-34a) in the examined organs in response to a viral stimulus and, on the other hand, inhibit (miR-21), which in both cases may be a determinant of the pathogenesis of RHD and tissue damage. Biomarkers of the Bax and Bax/Bcl-2 ratio promote more intense apoptosis after infection with the Lagovirus europaeus GI.2 genotype. Our findings demonstrate that miR-122 and miR-132 regulate oxidative stress in the pathogenesis of RHD, which is associated with tissue damage. The HO-1 biomarker in the course of rabbit hemorrhagic disease indicates oxidative tissue damage. Our findings show that miR-21, miR-16b, and miR-34a regulate three apoptosis pathways. Meanwhile, miR-122 and miR-132 are involved in two oxidative stress pathways.