In the U.S. the Supreme Court has ruled that the defendant must be
present at the start of their criminal trial in order for it to be
valid. Are there countries where the defendant has the right to
completely opt out of the process?
Frame change:
Conducting a criminal trial without the presence of the defendant at any stage of the process, which is called a trial in absentia is considered to be a grave violation of the rights of the criminal defendant in many, but not all, legal systems. A trial in absentia also calls into question the capacity of the criminal court involved to promptly impose its punishment upon a convicted criminal defendant if the defendant is convicted. Still, a minority of countries allow trial in absentia in some circumstances.
As explained by Wikipedia, "In some civil law legal systems, such as that of Italy, absentia is a recognized and accepted defensive strategy. Such trials may require the presence of the defendant's lawyer, depending on the country." As explained further at the same link:
Member states of the Council of Europe that are party to the European
Convention on Human Rights are bound to adhere to Article 6 of the
convention, which protects the right to a fair trial.
Trials in absentia are banned in some member states of the EU and
permitted in others, posing significant problems for the fluidity of
mutual recognition of these judicial judgments. The executing member
state possesses some degree of discretion and is not obliged to
execute a European Arrest Warrant if the country that is making the
request has already tried that person in absentia.
Conditions under which trials in absentia must be recognised include:
if the person can be said to have been aware of the trial; if a
counsellor took their place at the trial; if they do not request an
appeal in due time; and if they are to be offered an appeal. . . .
The Council of Europe has made commentary on judgments that are made
in absentia. The Committee of Ministers, in Resolution (75) 11, of 21
May 1975, stated that an individual must first be effectively served
with a summons prior to being tried. In this sense, the ministers are
emphasizing that it is not the presence of the accused at the hearing
that is of importance, rather the focus should be on whether or not
the individual was informed of the trial in time.
In a 1985 judgement in the case Colozza v Italy, the European Court
of Human Rights stressed that a person charged with a criminal offence
is entitled to take part in the hearings.
Several trials in absentia of U.S. criminal defendants, noted in the linked Wikipedia article, have been the subject of great diplomatic controversy and tension between the U.S. and Italy.
An arrest or at least nominal booking with the consent of the defendant is often considered the means by which a criminal court may obtain jurisdiction over the defendant to try the defendant. Assuring that right of a criminal defendant to be present at and participate in a trial of that criminal defendant has historically been a far greater concern in the law of criminal procedure than any right to be absent from the trial.
More generally, essentially all countries recognize that criminal defendants do not have an unrestricted right to be free from incarceration, after an arrest or after charges are filed against them, while a criminal case against them is pending but prior to a conviction or acquittal. Any release of a criminal defendant who has ever been booked or arrested, prior to the completion of a criminal trial and acquittal, is only allowed subject to terms and conditions set by the relevant law and the court in which the criminal case is pending. Some countries are more lenient than others about allowing criminal defendants to be at large prior to trial, but there is not a generalized unlimited right to be free prior to trial without conditions anywhere.
Any right to be absent from a criminal trial (which is almost always unwise because it deprives the lawyer for the criminal defendant from the assistance of the lawyer's client in conducting the defense, and because it shows a lack of respect for the people conducting the trial who will make the final decision on guilt and innocence and later, a sentence) is a subset of the general set of rules about when a criminal defendant can be at large prior to a conviction or acquittal.
As another answer notes, even in the U.S., sometimes a criminal defendant can be absent from a trial after appearing for the commencement of the trial, with the permission of the court. But, like pretrial release, this is only allowed in particular circumstances and subject to specific conditions. So, calling this a "right" of a criminal defendant, as opposed to an option of the court regarding how it conducts a criminal trial, is something of an overstatement in almost every case.