Everyone agreed that the decision of the College of Psychologists of Ontario impacted Peterson's freedom of expression. The College's decision itself said this. The question was whether the College conducted an "appropriate proportionately-focused balancing of Dr. Peterson’s right to freedom of expression and the statutory objectives of the College."
The reasons for judgment on judicial review are here: 2023 ONSC 4685.
The Ontario Superior Court of Justice was sitting as a divisional court conducting a judicial review of a decision made by the Inquiries, Complaints and Reports Committee of the College of Psychologists of Ontario.
To clarify the legal context, this was not a "trial" or an "appeal." There was first an administrative decision made by a body with a statutory mandate (the College). Peterson was not "taken to court" by the College. The College conducted its own investigation and on conclusion, issued an order that Peterson complete a remedial educational program. Such administrative decisions are reviewable by a process called "judicial review." In Ontario, these judicial reviews are almost always conducted by the Ontario Superior Court of Justice sitting with three judges as a divisional court.
Freedom of expression and reasonable limits
In Canada, Section 2(b) of the Canadian Charter of Rights and Freedoms says:
Everyone has the following fundamental freedoms: ... freedom of thought, belief, opinion and expression, including freedom of the press and other media of communication.
Those freedoms are subject to reasonable limits, the analysis of which takes on a different form depending on whether the limit comes from a law or from an administrative decision.
Section 1 of the Charter says:
The Canadian Charter of Rights and Freedoms guarantees the rights and freedoms set out in it subject only to such reasonable limits prescribed by law as can be demonstrably justified in a free and democratic society.
And the Supreme Court of Canada has drawn a parallel between a Section 1 analysis and a review of an administrative decision (Doré v. Barreau du Québec, 2012 SCC 12, para. 6):
If the law interferes with the right no more than is reasonably necessary to achieve the objectives, it will be found to be proportionate, and, therefore, a reasonable limit under s. 1. In assessing whether an adjudicated decision violates the Charter, however, we are engaged in balancing somewhat different but related considerations, namely, has the decision-maker disproportionately, and therefore unreasonably, limited a Charter right. In both cases, we are looking for whether there is an appropriate balance between rights and objectives, and the purpose of both exercises is to ensure that the rights at issue are not unreasonably limited.
The decision of the College of Psychologists
The Divisional Court summarized the College's decision:
 ... This Decision ordered Dr. Peterson, as a registered member of the College authorized to practice clinical psychology, to complete a specified continuing education or remedial program (a “SCERP”) regarding professionalism in public statements.
 The ICRC’s order followed an investigation into language used by Dr. Peterson in public statements earlier in 2022. In its Decision, the ICRC expressed its concern that Dr. Peterson’s comments may be “degrading, demeaning and unprofessional.” The ICRC concluded that some of the language used in Dr. Peterson’s public statements "may be reasonably regarded by members of the profession as disgraceful, dishonourable and/or unprofessional" and posed “moderate risks of harm to the public.” The risks of harm identified by the ICRC included “undermining public trust in the profession of psychology” and “may also raise questions about Dr. Peterson’s ability to appropriately carry out his responsibilities as a registered psychologist.”
The specific statements at issue are summarized later in the Divisional Court's reasons.
The Divisional Court also noted that the College's decision considered Peterson's right to freedom of expression. The College's decision said:
The Panel in no way disagrees that the Canadian Charter of Rights and Freedoms guarantees Dr. Peterson a right to freedom of expression. However, the Panel believes that as a Member of the College of Psychologists of Ontario, Dr. Peterson also owes a duty to the public and to the profession to conduct himself in a way that is consistent with professional standards and ethics. The Panel does not believe that Dr. Peterson's public statements are currently in line with professional standards and ethics. The proposed Undertaking would provide Dr. Peterson with the opportunity to better understand the standards and ethical expectations for regulated health professionals who make public statements of various kinds.
Judicial review of discretionary administrative decisions that impact Charter rights
Generally (absent special circumstances not present in this case), the role of the court conducting a judicial review of an administrative decision is to to determine whether the decision was reasonable.
The Divisional Court summarized the law:
 On an application for judicial review, therefore, the role of the Court is to ensure that the administrative decision-maker “proportionately” balanced the impact on Charter rights and the statutory objectives which “gives effect, as fully as possible to the Charter protections at stake given the particular statutory mandate”.
The Divisional Court accepted that the College's decision did engage Peterson's freedom of expression, but that this was a reasonable decision
The Divisional Court found that the College's decision was reasonable, and that it conducted the required balancing of Peterson's Charter rights against the statutory objectives guiding the College's discretion (paras. 38-67). The College's decision also met the more general requirements of reasonableness such as justification, intelligibility, and transparency (paras. 68-76).
Relevant to your headline question is paragraph 60 of the Divisional Court's reasons:
 In its correspondence with Dr. Peterson prior to the Decision, the Panel had agreed that Dr. Peterson’s right to freedom of expression was engaged, but that it was subject to his duty to the public and to the profession to conduct himself in a way that is consistent with professional standards and ethics.
Peterson has no further appeals as of right.
There may be an appeal to the Court of Appeal for Ontario, but, contrary to appeals of a typical final order of the Superior Court (Courts of Justice Act, s. 6(1)(b)), this would require leave (Courts of Justice Act, s. 6(1)(a)).
After that, there may be an appeal to the Supreme Court of Canada, also requiring leave.
Peterson's post-decision commentary — threshold applicability of the Charter and availability of judicial review
You have understood Peterson's post-decision commentary to be arguing that the College decision and judicial review are incorrect on the basis of:
the College being a governmental body, meaning they cannot regulate free speech the same way a private organization can.
And you ask in a comment:
I wonder about the legitimacy of Peterson's claim that such regulation of his free speech is unreasonable by virtue of the College being "fundamentally a governmental body".
The reason the Charter is relevant in the first place, and the reason judicial review of the College's decision is even available, is because the College is a governmental body. The College being a governmental body does not make the regulation of speech unreasonable; it makes such regulation and decision-making subject to the constraints of the Charter and oversight of the courts.
The analysis is not based on "agreement" or voluntary participation in the profession
Several comments (some deleted) have suggested that the analysis is as simple as something like "Peterson agreed to the oversight of the College by becoming a registered psychologist, so this isn't an infringement."
That would not be a correct analysis in Canadian law, and it isn't the analysis conducted by the Divisional Court.
Statutory regulatory bodies are creatures of government, so actions taken by such a statutory regulatory body, like the College, even over people who have voluntarily chosen to participate in the profession, might be found to disproportionately impact a member's Charter rights.
It is correct the profession of psychology in Ontario is "self-regulated" through the College, but this is simply another justification for deference to its decisions (Law Society of British Columbia v. Trinity Western University, 2018 SCC 32, para. 34). This fact does not render the decisions of the College free from Charter constraints or the requirements of proportionate balancing. There are decisions it could have made, or ways it could have made a decision, such that the decision would not be a proportionate balancing and would be quashed by a reviewing court on judicial review.