No, not directly. For that matter you can’t print STL directly either. Regardless of file format (STL, OBJ, AMF, 3MF, etc.), you’ll always have to run through a slicing program specific for the printer & material you are using to setup the job (akin to creating pst files for a 2D printer). Cura can process AMF files in addition to STL & OBJ.
3MF is not a Windows format, its a consortium (see www.3mf.io 2) to create the next generation format to replace STL. This won’t be commonplace until next gen printers start becoming ubiquitous that require an information rich format (i.e. the Multijet Fusion from HP). 3MF will enable voxel level control for multi-material, full fidelity printers. The consortia is made up for virtually every major player in the 3DP/AM space. But even when 3MF becomes standard, you’ll still need to process the file before taking it to print.