We provide, for the first time, robust observational constraints on the galaxy major merger fraction up to $zapprox 6$ using spectroscopic close pair counts. Deep Multi Unit Spectroscopic Explorer (MUSE) observations in the Hubble Ultra Deep Field (HUDF) and Hubble Deep Field South (HDF-S) are used to identify 113 secure close pairs of galaxies among a parent sample of 1801 galaxies spread over a large redshift range ($0.2<z<6$) and stellar masses ($10^7-10^{11} M_odot$), thus probing about 12 Gyr of galaxy evolution. Stellar masses are estimated from spectral energy distribution (SED) fitting over the extensive UV-to-NIR HST photometry available in these deep Hubble fields, adding Spitzer IRAC bands to better constrain masses for high-redshift ($zgeqslant 3$) galaxies. These stellar masses are used to isolate a sample of 54 major close pairs with a galaxy mass ratio limit of 1:6. Among this sample, 23 pairs are identified at high redshift ($zgeqslant 3$) through their Ly$alpha$ emission. The sample of major close pairs is divided into five redshift intervals in order to probe the evolution of the merger fraction with cosmic time. Our estimates are in very good agreement with previous close pair counts with a constant increase of the merger fraction up to $zapprox 3$ where it reaches a maximum of 20%. At higher redshift, we show that the fraction slowly decreases down to about 10% at $zapprox6$. The sample is further divided into two ranges of stellar masses using either a constant separation limit of $10^{9.5} M_odot$ or the median value of stellar mass computed in each redshift bin. Overall, the major close pair fraction for low-mass and massive galaxies follows the same trend. These new, homogeneous, and robust estimates of the major merger fraction since $zapprox6$ are in good agreement with recent predictions of cosmological numerical simulations.
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