Real suffered back-to-back setbacks, first losing 3-2 to Barcelona in Sunday's Spanish Super Cup final, then crashing out of the Copa del Rey on Wednesday after a 3-2 defeat to second-tier Albacete, a result that also marked Arbeloa's first game in charge following the departure of Xabi Alonso.
Real sit second with 45 points from 19 matches, four fewer than leaders Barcelona.
"I have a lot of respect for the opinion of the Bernabeu. I understand that the fans are hurt and disappointed, but I'm going to ask them to support their players," Arbeloa told reporters on Friday.
"Great achievements have been made when the Bernabeu has stood by its players. I'm going to ask them to stand by us, support us and hopefully this season will end the way we want it to."
Arbeloa, a two-time Champions League winner as a player for Real, said he would not change anything about the defeat at Albacete.
"I understand that people are looking for someone to blame, but I work to find solutions. When things aren't going well, it's because I need to help my players better," he added.
The 42-year-old also recounted a story he shared with the dressing room about the mindset required.
"I told them a story in the dressing room about how it took me many years to win a European Cup. I was world champion, European champion, I had won a lot ... And when I finally won that European Cup...," the Spaniard said.
"I got on the bus and there was a teammate behind me who had been there for a year and he said to me: let's go for another one, shall we? I said to the dressing room: Do you know who he is? Your captain (Dani Carvajal), who was already thinking about another one. The past doesn't matter."
In a major boost, Arbeloa said forward Kylian Mbappe, LaLiga's leading scorer this season with 18 goals, is fit and will rejoin the squad, having recovered from a sprain in his left knee. However, winger Rodrygo will not be available.