Members of the Communication Workers Union (CWU) voted to 4-1 in favour of the walkout on November 4.
The management of Royal Mail, which listed on the stock market last week, expected a vote in favour of strike action and criticised the prospect of a disruption to services in the run-up to Christmas, its busiest time.
The CWU sent ballot papers to over 115,000 Royal Mail staff and said 78% had voted to strike, on a turnout of 63%.
The union said further action was under consideration and added that it planned to carry out a second ballot asking workers to support a boycott of the handling of competitors' mail, which Royal Mail delivers on their behalf.
The development comes after Royal Mail staff were given free shares in the company, in an attempt to improve labour relations.
On Tuesday, Royal Mail confirmed that full-time staff would each receive 725 shares, worth £3545 at Tuesday's close.
At market close last night the shares were at 475p - below their 489p high, but well above their sale price of 330p.
The CWU, which fiercely opposed privatisation over fears it would lead to poorer job conditions and services, in July rejected Royal Mail's offer of an 8.6% pay increase over three years, criticising proposed changes to pensions.
It had already accused the Government of "deliberately creating a scramble" for shares, leading to many private investors cashing in their allocation at a big profit in the past few days.
CWU deputy general secretary Dave Ward said: "Postal workers have spoken very clearly - they care about their jobs, terms and conditions far more than they care about shares.
"The stakes have become much higher for postal workers since privatisation, making this ballot more important than ever. Postal workers will not be the people who pay for the profits of private operators and faceless shareholders."
Mr Ward said the question now was whether a privatised Royal Mail still wanted an agreement.
"We have offered the company a two-week period to reach an agreement and, having already had many hours of negotiation, this is achievable if there is a will. The clock is ticking for both sides and we need Royal Mail to work to reach agreement before this deadline.
"What we want is a ground-breaking, long-term agreement that not only protects postal workers' job security, pay and pensions, but will also determine the strategy, principles and values of how the Royal Mail Group will operate as a private entity."
The Direct Marketing Association, representing the UK advertising mail industry - which it says accounts for £1 billion of Royal Mail's £9bn turnover - said strike action would have a severe impact on customers.
"The build-up to Christmas is a critical period that typically accounts for more than half of businesses' annual revenues ... Any disruption to service would quickly lead businesses to take their custom elsewhere," DMA's executive director Chris Combemale said.
Business Secretary Vince Cable has been asked to give fresh evidence to a committee of MPs over the privatisation following an appearance last week, when he faced tough questioning about the sell-off and the price set for the share issue.
Investment bank Lazard is also set to be questioned next month by the Business Select Committee over the pricing of Royal Mail and concerns that institutional investors were allowed to buy into the company too cheaply.