Prime minister David Cameron has called for a Britain where "no matter where you come from, what god you worship, the colour of your skin, what community you belong to", people are able to reach the top of their chosen professions, Professor White writes.
But the academic says that this "ladder philosophy" allows some people to progress up the social scale while - provided that there are still plenty of good jobs for children from wealthier families - doing nothing to threaten the existing social order.
However, he added, as well-paid jobs for graduates become increasingly scarce, it is people "less well-cushioned by parental wealth and connections" who are "likely to lose out".
"Better-off families are likely to do what they can to ensure that their children end up at the top end of the hourglass, and do not slip down towards the bottom," Professor White says. "We can expect an intensification of stratagems to give them favourable treatment in the examination stakes."
It is the exam system - rather than exams themselves - that is particularly at fault, he claims. This system gives higher priority to the more abstract and theoretical subjects, in which pupils from educated backgrounds have a natural advantage. It also allows wealthy parents to move to favourable catchment areas, buy in additional coaching or send their children to private schools.
"But, if all of these and other such advantages were done away with, would there still be any point in retaining the examination at all?" Professor White writes. "It would, by definition, no longer have its old rationale as an instrument of class power.
"Examinations.are not necessary to motivate students to learn, to provide a record of their work, to enable universities and employers to select entrants, to show how well a school is doing."
In their place, Professor White recommends introducing an ongoing record of achievement, or student profile, which records pupils' progress throughout their school career. This could be used to collate information not only about children's formal knowledge but also about their practical and personal skills.
"Employers should be able to search web-based applications for the qualities and achievements they are looking for," he says. "Since many, perhaps most, employers are looking for more than academic qualifications, they should welcome the possibility to trawl for skills, enterprise and other personal qualities of interest."
But Tim Oates, a senior official at Cambridge Assessment, which oversees the OCR exam board, described Professor White's argument as "extremely confused".
"I can use a kitchen knife to prepare fine food or I can use it to kill my partner," he said. "The fact that kitchen knives are used in violent acts against people doesn't mean that we should get rid of kitchen knives.
"I'm not naively saying that public examinations aren't value-laden. But examinations provide a public institution of awarding, which is amenable to public scrutiny. It's essential that we have assessment, because it takes it away from arbitrary judgements, laden with prejudice. It's precisely these sorts of judgements that prevent social mobility."
Dame Joan McVittie, headteacher of Woodside High School in North London, pointed out that schools already kept a record of pupils' non-academic skills and qualities, such as teamwork and confidence.
"It's absolute nonsense to say that we should abolish examinations completely," she said. "They allow you to hold schools to account and quality-assure the education that they're giving. I don't think we're ready for this egalitarian society he's proposing."
Who Needs Examinations? is published by Institute of Education Press, priced at pound;14.99