The government says it is working to "eliminate" child poverty and to improve life chances for children.
A huge difference in healthy lifestyles was uncovered by the research.
Of the 37 countries studied, children from the poorest and richest families in the UK showed the biggest difference in the levels of healthy eating.
Children from the poorest households were found to be less physically active than their richer peers.
Unicef also found a big gap in the levels of education in Britain.
In reading, maths and science, the UK lags behind Slovenia, Poland and Romania.
Overall, the UK places 25th out of 37 countries in terms of education inequality.
And if it weren't for "social transfers" such as working tax credits, the UK would have an income gap between rich and poor among the highest in Europe.
The report found that overall, the UK performs better on income inequality, where it places 7th out of the 37 countries.
Poor British children were also found to be less satisfied with their lives than their better off peers.
Inequality in life satisfaction found the UK ranked 20th out of 35 EU countries.
Income inequality is very, very big and growing. Lily Caprani, Unicef
Unicef UK's Lily Caprani said: "We must be more ambitious for our children. Britain can and must do better.
"What really stands out for the UK is that children who are at the very bottom of the income distribution, so those with the most disadvantages, are doing particularly badly when it comes to things like healthy eating, because we are really slipping behind other nations.
"Income inequality is very, very big and growing.
"For those children in the most disadvantaged households, if it wasn't for things like tax credits to protect their incomes, we'd actually see the biggest levels of inequality in the entire industrialised world."
Matthew Reed of The Children's Society, said: "The finding that poorer British children feel less satisfied with their lives than their richer classmates – and less happy than young people in other countries – is deeply concerning but sadly not surprising."
We're working to eliminate child poverty and improving life chances for children UK Government
A government spokesman said: "We're working to eliminate child poverty and improving life chances for children and there are now 300,000 fewer children in poverty.
"We know there's more we can do and that's why we've introduced the National Living Wage, which is increasing the incomes of the lowest paid.
"All infant pupils can now get free school meals - meaning 1.3 million more children get a nutritious free meal at lunchtime, saving families hundreds of pounds."