Using the approach formulated in the previous papers of the author, a consistent procedure is developed for calculating non-classical asymptotic power terms in the total correlation function hc(r) of a “critical” fluid. Analyzing the Ornstein-Zernike equation with taking account of the contribution ˜hc4(r) in the direct correlation function cc(r) allows us to find, for the first time, the values of transcendental exponents n′=1.73494... and n″=2.26989...which determine the asymptotic terms next to the leading one in hc(r). It is shown that already the simplest approximation based on only two asymptotic terms, ˜r−6/5(it was found earlier) and ˜r−n′, leads to the functions hc(r) and cc(r), which agree (at least, qualitatively) with the corresponding correlation functions of the Lennard-Jones fluid (argon) in the near-critical state. The obtained results open a way for consistent theoretical interpretation of the experimental data on the critical characteristics of real substances. Both the theoretical arguments and analysis of published data on the experimentally measured critical exponents of real fluids lead to the conclusion that the known assumption of the similarity of the critical characteristics of the Ising model and the fluid in the vicinity of critical point (the “universality hypothesis”) should be questioned.