You use the words 'without proof of wrong-doing'.
There isn't a need to 'prove' anything in these situations. The proof required in criminal convictions is 'beyond reasonable doubt' but in civil law (and employment law is really a sub-set of that) the burden of proof would be a balance of probabilities.
What actually happens is that the employer, having suspended the employee' should conduct a reasonable investigation, using a reasonable process (which should include inviting the employee to an interview to hear his/her side of the story) and come to a conclusion as to the gravity of the alleged misconduct. There isn't a need to 'prove' anything - if the manager conducting the investigation concludes that from the evidence provided that it believes that the misconduct has taken place, the employer can decide what to do in terms of sanctions (There is normally a review of the decision process first).
If the misconduct is considered extreme, it may decide the conduct constitutes gross misconduct for which a possible sanction is dismissal without notice. Dismissal for (simple) misconduct, with paid notice, is perhaps quite rare - final written warning is more likely.
If the dismissed employee feels aggrieved he/she can always appeal - firstly to the employer, then secondly to an Employment Tribunal.