The International Space Station Intergovernmental Agreement covers the ISS. Under the IGA, as it is often known, the laws of the nation (or region, in the case of the European Union) apply to its citizens and property :
This extension of national jurisdiction determines what laws are
applicable for activities occurring on a Partner’s Space Station
elements (e.g. European law in the European Columbus Laboratory). This
legal regime recognises the jurisdiction of the Partner States’s
courts and allows the application of national laws in such areas as
criminal matters, liability issues, and protection of intellectual
property rights. Any conflicts of jurisdiction between the Partners
may be resolved through the application of other rules and procedures
already developed nationally and internationally.
The basic rule is that 'each partner shall retain jurisdiction and
control over the elements it registers and over personnel in or on the
Space Station who are its nationals' (Article 5 of the
Intergovernmental Agreement).
So if you are American, US law applies wherever you are in the station. If you are French, European Union law applies to you. If an American and a Frenchman get into a fight, during which Russian property is damaged, things get extremely complicated. To handle this, there is a cross-waiver of liability. Under it, liability for all damage is waived, including:
(i) Bodily injury to, or other impairment of health of, or death of,
any person;
(ii) Damage to, loss of, or loss of use of any property;
(iii) Loss of revenue or profits; or
(iv) Other direct, indirect, or
consequential damage.
So that covers everything serious, as long as it happened as part of official duties. Obviously there is a big grey area there. In the above example, if there was some shouting and shoving while on duty and someone accidentally kicked a laptop monitor, perhaps it would be permitted to come under the waiver. If it is decided that an act was criminal, then
In a case involving misconduct on orbit that: (a) affects the life or
safety of a national of another Partner State or (b) occurs in or on
or causes damage to the flight element of another Partner State, the
Partner State whose national is the alleged perpetrator shall, at the
request of any affected Partner State, consult with such State
concerning their respective prosecutorial interests. An affected
Partner State may, following such consultation, exercise criminal
jurisdiction over the alleged perpetrator provided that, within 90
days of the date of such consultation or within such other period as
may be mutually agreed, the Partner State whose national is the
alleged perpetrator either:
(1) concurs in such exercise of criminal jurisdiction, or
(2) fails to provide assurances that it will submit the case to its
competent authorities for the purpose of prosecution.
That quote comes from the document itself. After that point it starts talking about extradition procedures. So it is a mess and let's hope it never happens on the ISS. If a citizen of Partner State A beat the hell out of a citizen of Partner State B, and Partner State B was not pleased by how Partner State A was handling the matter, well then they could demand extradition and apply their own laws.
Bottom line, everything up there depends on diplomacy.