Despite numerous studies of water structures at the two-dimensional water-solid interfaces, much less is known about the phase behaviors of water at the one-dimensional (1D) liquid-solid interface. In this work, the 1D interfacial water phase behavior on the outer surface of carbon nanotube-like (CNT-like) models is studied by tuning the Lennard-Jones potential parameter ε of the surface atoms at various temperatures. Extensive molecular dynamics simulations show that ice nanotubes (INTs) can be spontaneously formed on CNT-like model surfaces without nanoconfinement. INTs with flat-square walls (INTs-FSW) are formed on the CNT-like model surface when the ε value is beyond a critical value εc. The value of εc exponentially increases as temperature rises. Contrary to the prevailing formation of INTs-FSW at a relatively strong water-surface interaction, INTs with bilayer hexagonal walls are formed at a weak interfacial interaction with the ε value being in a modest range. An ε-T phase diagram is constructed for the 1D interfacial water on the CNT(100, 0) model surface. Rich phases of H2O are given in different regions of the phase diagram, depending on the water-surface interaction. This comprehensive study not only provides new insight into the phase behavior of 1D interfacial water but also can guide future experiments to produce INTs without nanoconfinement.