According to a report by Us Weekly, Artem responded Friday to Nikki's divorce filing ... and he's asking for joint legal and physical custody of their 4-year-old son, Matteo.

The "Dancing With The Stars" pro is also asking for spousal support from Nikki.

TMZ broke the story ... Nikki is asking for sole legal and physical custody over Matteo ... and in her divorce filing, she checked the box to block the court's ability to award spousal support to either her or Artem.

Based on Artem's new filing, it seems the estranged couple is heading for a fight over child custody and spousal support issues.

There are two things Artem and Nikki agree on in their divorce filings ... they both cite irreconcilable differences as the reason for the split and list August 29 as the date of separation, the day Artem was arrested for felony corporal injury to a spouse or cohabitant.

As we told you ... Artem originally called 911 on Aug. 29 to request medical assistance for the alleged victim ... only to call off paramedics minutes later. Police still showed up and arrested him.

Artem has not been formally charged yet ... though we're told investigators last week turned the case over to prosecutors.