an official process to discover the facts about something bad that has happened: (-- Cambridge Online)
The natural result of an inquiry (as the word is generally used in the UK) is a report, normally intended for publication. So yes, the report is what is published, rather than the inquiry. It is likely to contain details of evidence considered, conclusions drawn (with reasons), and a summary. With respect to Public Inquiries, the process is outlined by wikipedia:
The conclusions of the inquiry are delivered in the form of a written report, given first to the government, and soon after published to the public. The report will generally make recommendations to improve the quality of government or management of public organisations in the future.
You will commonly see phrases such as "demands to publish the Chilcott Inquiry report"; you're unlikely to see the word 'report' omitted, and this shouldn't be confused with "demands for an Inquiry".
However, and further to my earlier comment, the phrase 'publish an inquiry' will be commonly understood to mean 'publish the final report produced by an inquiry process', in my opinion. Personally, I'm happy with 'publish the results of an inquiry'.
You might bear in mind possible confusion where an interim statement or report is produced, and also where a meta inquiry into the workings of a previous inquiry might result in a 'report on an inquiry' (as suggested in the question), as opposed to the 'report of or by an inquiry.