The additional staff would be recruited using money saved from ending universal free prescriptions, which were introduced by the SNP government almost three years ago.
Exemptions would be maintained for children, students aged up to 19, over 60s, pregnant women and people on income support and jobseekers allowance, meaning around half the population would not face charges.
Prescriptions became free to all in April 2011 after a phased reduction in charges, with the cost of funding the policy standing at around £57million for 2011/12.
The Tories say the 1000 extra nurses and midwives would cost £30-£36million a year.
Miss Davidson said: "Under the SNP, the number of nurses and midwives in Scotland has gone up and down like a fiddler's elbow. Two thousand posts gone over two years.
"Right now - hundreds of places down, creating an intolerable pressure on those who are left.
"Millions spent on bank or agency nurses to plug the gaps. It's not good enough. It's not good enough for staff and it's not good enough for patients."
She added: "It's time we did something about it. That's why today I am able to announce the Scottish Conservatives will pledge an extra 1000 nurses and midwives for Scotland. And, once introduced, we will not let numbers drop below that mark.
Health Secretary Alex Neil said: "It is not a case of funding either prescriptions or nurses. In Scotland's NHS we can and will provide free prescriptions, pay our NHS staff a fair wage and protect an NHS free at the point of need."