We investigate the effects of inverse Compton scattering by electrons and positrons in the unshocked winds of rotationally-powered binary pulsars. This process can scatter low energy target photons to produce gamma rays with energies from MeV to TeV. The binary radio pulsars PSR B1259 - 63 and PSR J0045 - 73 are both in close eccentric orbits around bright main sequence stars which provide a huge density of low energy target photons. The inverse Compton scattering process transfers momentum from the pulsar wind to the scattered photons, and therefore provides a drag which tends to decelerate the pulsar wind. We present detailed calculations of the dynamics of a pulsar wind which is undergoing inverse Compton scattering, showing that the deceleration of the wind of PSR B1259 - 63 due to 'inverse Compton drag' is small, but that this process may confine the wind of PSR J0045 - 73 before it attains pressure balance with the outflow of its companion star. We calculate the spectra and light curves of the resulting inverse Compton emission from PSR B1259 - 63 and show that if the size of the pulsar wind nebula is comparable to the binary separation, then the gamma-ray emission from the unshocked wind may be detectable by atmospheric Cherenkov detectors or by the new generation of satellite-borne gamma-ray detectors such as INTEGRAL and GLAST. This mechanism may therefore provide a direct probe of the freely-expanding regions of pulsar winds, previously thought to be invisible. (C) 2000 Elsevier Science B.V. All rights reserved. PACS. 95.30.Jx: 95.55.Ka; 97.60.Gb; 98.70.Rz.