"Cerere incidenta prin care una dintre parti urmareste, prin introducerea in proces a unei a treia persoane, sa fie aparata sau despagubita de catre aceasta."
http://www.rubinian.com/dictionar_detalii.php?id=2589

So "chemare in garantie" does seem to have the same meaning as the French "appel en garantie", which is "to introduce/bring in a third party" (to a case):

http://www.granddictionnaire.com/ (I'm afraid you'll have to do the search yourself, it won't let me give you the exact page!)

To me this sounds rather like "to make a third-party claim":

"THIRD-PARTY CLAIM - A claim by the civil defendant/respondent against a party not already named in a court or other adversarial proceeding."
http://www.lectlaw.com/def2/t026.htm

"Third Party Claim: A claim filed by a defendant who feels that some third party (not yet involved in the case) is responsible and owes him or her money that should be used to satisfy any judgment the plaintiff may win from the defendant."
http://www.peoples-law.org/finding/glossary/def-fam-law.htm

In the sentence in which you have it, though, since it is not referring to a specific claim (just the idea of doing this in general) I would pluralise it and say "to make claims against third parties/to make third-party claims". In the US I believe it would be "to file" rather than "to make", e.g.

"Your claim is served to the respondent, who is given time to provide an answer. After the respondent is served, both parties are then responsible for providing--or serving--copies of all other documents, pleadings, correspondence, etc., directly to the other parties and for providing additional copies of any documents to NASD for its record and for the arbitrators. Keep in mind that the respondent may also *file a claim against a third party*, or may file a counter claim against you.
http://www.nasd.com/web/idcplg?IdcService=SS_GET_PAGE&nodeId...