Police blocked access to a main road in the capital and used pepper spray to try to control the crowd.
Protesters threw punches and demanded respect from police.
Venezuela is facing a shortage of many basic items, including medicines and medical equipment.
Many feel that elderly people have been particularly affected by the crisis in the health system.
In another development, Mr Maduro has sacked his health minister after she released official figures showing a sharp rise in child mortality and maternal deaths over the past two years.
Antonieta Caporale, a gynaecologist, had been in the post for only four months and was replaced by the Vice-Minister of Hospitals, Luis López.
Riot police used pepper spray to try to disperse the crowd
An elderly opposition protester attacks the security forces
"I'm here to defend my grandchildren, to defend my country," 78-year-old Rafael Colmenares told Reuters news agency during the demonstration in Caracas.
"Respect the elderly," many demonstrators shouted.
Merida too saw protests
The march was diverted but it eventually reached the office of Venezuela's ombudsman, where the protesters accused the security forces of brutal suppression.
At least 39 people have been killed since the current wave of protests began six weeks ago.
The unrest was triggered by a Supreme Court attempt to take over powers from the opposition-controlled National Assembly on 29 March.
It reversed its decision a few days later, but by then the opposition had seized the momentum.
There have been almost daily protests in Venezuela since the beginning of April
President Maduro's term ends in January 2019
To maintain pressure on President Maduro, opposition leaders have begun organising themed marches.
Last Friday, they urged women to take to the streets. There was a candle-lit march. And Friday's protest was dubbed "The Grandparents March".
The government responded with a rival march to show support to Mr Maduro, with thousands of pro-government elderly people taking to the streets of Caracas.
The opposition is calling for fresh presidential elections.
Mr Maduro says they are trying to create unrest to unsettle his democratically-elected socialist government and seize power.