It is a difficult quest to arrive at a Utopia of growing, sovereign nations, so maybe it is easier to ask what it would take to maintain such a state of the world, before wondering how to get there.
Consider how a single nation maintains peace within itself; any peaceful modern nation is based on some fundamental ideas - a legal system and some form of representational government, whether secular or religious. The legal system comprises a set of laws that are applied without preference or prejudice to the whole of the population, patrolled by a police force and enforced with by an independent judiciary. The application of the legal system is independent of the political system, even though the later may add to the body of law that is enforced. The political system represents the people in some way, and controls an executive that performs the day to day running of the government. The jurisprudence might by driven by a civil code, case history, or a religious code.
At the supranational level, we can identify similar bodies today. At the continental level the European union provides a functionaing model for co-operating nations, and has fairly well established parliament providing elected representation, a court system, a legislative body, an executive under the high level guidance of the European Council.
On the global scale, we have the United Nations, comprised of a representative body, an executive, a court system, the Security Council responsable for maintaining international peace, and the Economic and Social Council. Independent of the UN, we have the International Criminal Court and Interpol, the World Bank and the International Monetary Fund.
How well are these institutions working, and are they capable of maintaining peace in Utopia?