Yes, the orbit of Mercury can be explained using special relativity alone, without invoking classical mechanics. In fact, one of the early triumphs of Albert Einstein's theory of special relativity was its ability to account for a long-standing anomaly in Mercury's orbit.
Classical mechanics, based on Newton's laws of motion and gravitation, predicts that the orbit of a planet should remain constant over time. However, careful observations of Mercury's orbit revealed a small but noticeable discrepancy known as the precession of the perihelion. The perihelion is the point in an orbit closest to the Sun, and the precession refers to the slow rotation of this point around the Sun over time.
Einstein's theory of special relativity, which he formulated in 1905, introduced a new understanding of gravity and the nature of spacetime. According to special relativity, mass and energy can curve spacetime, and objects moving through this curved spacetime experience gravitational effects.
Einstein's field equations of general relativity, developed later in 1915, provided a more precise and comprehensive description of gravity. Using these equations, it is possible to calculate the precise precession of Mercury's orbit and obtain results that match the observed data. The relativistic corrections account for the additional rotation of the perihelion, which cannot be explained by classical mechanics alone.
Therefore, special relativity, with its extension into general relativity, offers a more accurate explanation of Mercury's orbit, including its precession, compared to classical mechanics.