The Laborers International Union of North America (LiUNA) advanced the proposal as part of a campaign to get 50 public companies to do the same.
"We really just want to know that they have a plan in place, that they're developing internal candidates, that they have an emergency plan in place and that they're really working on using the business strategy of the company to develop a future successor," Jennifer O'Dell, assistant director of the union's corporate affairs department, said.
O'Dell named Intel and HP, both neighbors of Apple in Silicon Valley, as corporations it convinced to adopt such a plan.
Opponents said Apple is setting revenue and profit records. It is putting a dent in PC sales by the success of the iPad, so it is right on course.
"I trust Steve Jobs; I trust his executive team, and I think it's really a lame idea for these little stockholders to make so much noise," succession proposal opponent Larry Fritzlan said.
Apple shareholder Tim Shaw was on the losing side.
"Having confidence in who might come in and take his place or the system in selecting that individual, I think, is important to a lot of shareholders," Shaw said.
Professor Ron Gilson is an expert on corporate governance at Stanford Law School. He says succession is a board of directors issue.
"It is important that the board take responsibility for making sure that succession plans is part of the regular process," Gilson said.
The union says it will try to meet with Apple to resolve the issue.
Not a single shareholder asked Apple executives running the meeting about Jobs' condition. Two shareholders did express best wishes for his recovery. RadarOnline.com posted video several days ago showing Jobs unsteady on his feet as he left a building and got into the passenger side of a parked car.
Apple chief operating officer Tim Cook, who has assumed Jobs' duties on an interim basis, fielded questions from shareholders after the 23-minute formal meeting concluded during which the slate of seven directors, including former Vice President Al Gore, was re-elected. Cook was asked about how Apple is meeting growing competition from Google's Android operating system, worker safety at outsourced manufacturing plants in China, and progress on a new data center under construction in North Carolina.
Apple is having an event next Wednesday in San Francisco where it is expected to announce the iPad 2.