International organisations are products of international law, as they are invariably created by means of multilateral treaties. It is therefore not surprising that the International Court of Justice (“ICJ” or “World Court”) has made it quite clear on several occasions that international organisations “are bound by any obligations incumbent upon them under general rules of international law” (Advisory Opinion on the Interpretation of the Agreement of March 25th, 1951 between WHO and Egypt  I.C.J. Reps at 89-90. See also the Reparation for Injuries Case  I.C.J. Reps 174 at 179: an international organisation is “a subject of international law and capable of possessing international rights and duties”).
It is now generally accepted that international organisations, as international persons, are bound by the customary rules of international law, including the customary aspects of international human rights law. By way of example, Sands and Klein, editing Bowett’s Law of International Institutions, take the view at page 456 that:
“notwithstanding the fact that an international organisation is not a party to, say, a human rights treaty …if the rule contained in an agreement is reflected in customary international law then it can, as such, bind an international organisation. It has been suggested, for example, that the World Bank is not subject to general international norms for the protection of fundamental human rights. In our view that conclusion is without merit, on legal or policy grounds.”
The right to a fair and, indeed, a public hearing by an independent and impartial tribunal in the determination of rights and obligations is now undoubtedly a general rule of international law; it is a ‘fundamental human right’ that finds its origins in article 10 of the Universal Declaration of Human Rights (“the Declaration”). Building upon the Declaration, which was proclaimed by the General Assembly of the United Nations on 10th December 1948, the states of Europe sought in 1950 to enforce certain Declaration rights by agreeing the European Convention on Human Rights (“ECHR”). Article 6(1) of the ECHR provides that in the determination of civil rights and obligations “everyone is entitled to a fair and public hearing within a reasonable time by an independent and impartial tribunal established by law.”
The geographical limitation of the ECHR was to a certain extent addressed and remedied by the United Nations opening for state signature on 19th December 1966 the International Covenant on Civil and Political Rights (“ICCPR”), which came into force on 23rd March 1976. In conformity with article 10 of the Declaration and article 6 of the ECHR, article 14(1) of the ICCPR states that in the “determination … of his rights and obligations in a suit at law, everyone shall be entitled to a fair and public hearing by a competent, independent and impartial tribunal established by law.” Similar provisions can be found in other international human rights treaties, such as article 8 of the 1969 American Convention on Human Rights and articles 10 and 20 of the 2012 ASEAN Human Rights Declaration.
Most, if not all international organisations enjoy as a matter of law some form of immunity from legal suit in their respective member states. This means that international civil servants, as the employees of international organisations, are prevented from pursing employment related complaints in national employment courts and tribunals. In order to maintain their immunity from suit, international organisations are thus obliged to provide their employees with a forum in which they can litigate their disputes and, as such, international organisations have over the years developed and implemented various forms of internal justice mechanisms. But merely providing an international civil servant with a mechanism by which his or her case can be heard is insufficient, as international law, as is clearly demonstrated above, requires that the hearing must be public and above all fair, which means that the mechanism must offer all the guarantees inherent in the notion of due process. The consequences for an international organisation of failing to provide for a fair internal system of justice is that it risks losing its immunity and therefore risks being sued successfully in national courts. In Waite and Kennedy  116 ILR 121, the European Court of Human Rights (“ECtHR”), dealing with a case concerning the European Space Agency, made plain that “a material factor in determining whether granting […] immunity from […] jurisdiction is permissible is whether the applicants had available to them reasonable alternative means to protect effectively their rights under the [ECHR].”
The right to the enjoyment and protection of human rights is not lost merely because an individual decides to work for an international organisation. Human rights ‘black-holes’ do not exist and they most certainly do not exist within organisations created and maintained by states. So, in short, the answer to the question is yes, international civil servants, as the employees of international organisations, have the right to have their employment complaints heard and heard fairly.