Short answer, no. Before the 'fall' of Rome, the empire generally had bigger issues to worry about than to spread Christianity by force, and even if it could, theologians such as Augustine argued that forced conversions were in fact bad. There is also the problem of what brand of Christianity they should spread, since both Constantius II and Valens were Arians, whilst others such as Theodosius were Nicene Christians. Some, like Valentinian I, wanted to steer clear of religious controversy altogether. Missionaries however did try to spread the faith, and the Goths were converted to Arianism by Ulfilas, whilst many other Germanic tribes adopted Arianism as their faith seemingly voluntarily (or just because we don't have accounts of their conversion).

The same was true after the 'fall' of the West. Though it has to be noted that many of the Eastern Roman Empire's immediate neighbours were Christians already - the Armenians were miaphysite Christians, some of the Arab tribes had also converted, whilst the borderlands with Persia were full of Nestorian Christians. Again, missionaries were sent to Ethiopia and to the Caucasus Mountains and beyond, and we have several accounts of Lazi and Hunnic royalty being converted, but they were not pressured by military force - if anything, it was Eastern Roman policy to get foreign leaders to convert so that they would open a second front against Persia! Again, there were issues of theology and unclear policy ever under able emperors such as Justinian. For example, Ethiopia was allegedly the recipient of two missions, one miaphysite and one Chalcedonian, apparently because Justinian and the empress Theodora were at loggerheads theologically too, which confused Eastern Roman policy further. This continued into the seventh century too, and although the Eastern Romans were all too happy to massacre or punish 'heretics' within their own territories, they never tried to impose their faith by force on foreigners. Interestingly though, Justin II on one occasion invaded Persia to help the Christian Armenians (even though they were miaphysites), suggesting that religion did become more important in terms of foreign policy, but it was not enough for Roman emperors to wage wars of faith against their neighbours. I'm not too familiar with Byzantine policies after the seventh century, so I'll let someone else answer that :)