Hmm, a lot of European colonialism was spurred on by rivalries between the colonial powers. If this doesn't exist, I'm not sure much colonisation would go on. Unless the federal nature of this United Europe allows different regions to still have their own colonies, or there's a powerful North African/Arabian state, or a China or Japan etc with a huge reach, there might not be much effort put in. Maybe the settlement would be akin to OTL Britain's colonisation of Australia, allowing much greater strength for the Native American nations, and also encouraging the development of new cultures from the settlers (as in South Africa OTL).
This is assuming the nation would be genuinely relatively united, somehow. Roman Empire never dies? Random hyper-zealousness brings everything together under the Holy Rule of Constantinople/Rome/Moscow/Jerusalem? Everybody outside San Marino dies, allowing it to colonise the entire landmass?