You can "measure" impedance through simulation relative to coaxial probe on a Smith chart or using complex value of reflection coefficient. Be aware that good return loss also may occur with poor matching in very narrow band area, especially if your simulation is done only for one frequency point. You can check it by increasing feeding line lengths by quarter wavelength. In many papers I've seen matching between free-space and antenna is done using dielectric slabs or perforated substrate. It is not your case unless you are using some foams/superstrate/dielectric with gradually increased perforation. It is worth noting that even planar antennas may have input impedance below 50 Ohm.
In other words,
1) ensure that matching is good enough for your bandwidth, not single frequency point.
2) no, you do not need to match for 370 Ohms