The demand coincides with fresh fears about possible EU-wide health outbreaks this coming winter.
A report published on Thursday said that EU-wide planning to protect citizens from serious health threats such as pandemic influenza suffers from "significant weaknesses."
The report was drafted by the European Court of Auditors (ECA), the EU's financial watchdog, and released in Brussels on Thursday.
It says that although a number of important steps have been taken in recent years, member states and their public health authorities "still need to work together better."
EU member states have the principal responsibility in the area of public health and the European Commission's role consists mainly of providing support and taking complementary action.
But the ECA points out that the EU has identified serious cross-border threats to health as an area where member states can act more effectively together.
The EU decision on serious cross-border threats to health in 2013 introduced important changes to advance planning and response coordination, says the ECA.
It also strengthened the health security committee set up informally by EU health ministers in 2001.
The auditors conclude that although the 2013 decision was an important step to improve the EU's health security framework and better prepare the EU to deal with serious threats to health, "significant weaknesses" at member state and commission level remain.
Although the health security committee has proved to be very important, it faces "strategic and operational challenges that need to be resolved."
Speaking in Brussels at the launch, Janusz Wojciechowski, the member of the European Court of Auditors responsible for the report, said, "Increased travel and trade allow diseases to spread rapidly across borders, which means health security in one member state often depends on that of its neighbours.
"More needs to be done to address these weaknesses in planning and coordination if EU citizens are to benefit fully from what has been set up so far."
The auditors found delays in implementing and developing the 2013 decision.
Coordination of preparedness planning was improved, but the procedures need to be more robust and better defined.
For example, they say member states have not sufficiently speeded up their joint purchase of pandemic influenza vaccines and there is no EU-wide system to address urgent needs for vaccines or other medical countermeasures.
The report says, "The existing systems for early warning and response and epidemic surveillance have been operational for several years and their important role at EU level is widely recognised."
However, there is scope to upgrade the early warning and response system. The latest updates to procedures for dealing with serious chemical and environmental threats have not yet been tested.
The auditors also found weaknesses in the performance of the EU's health programme for protecting citizens from health threats.
They say, "Most of the audited health threat activities between 2008 and 2013 showed a lack of sustainable results, despite performing well in terms of producing their agreed deliverables. This limited their contribution to protecting citizens from health threats."
They also found weaknesses in measuring the health threat objective for the period 2014 to 2020 and a relatively low level of spending between 2014 and 2016.
The auditors identified a number of gaps in the Commission's internal coordination of health security activities across different programmes and services.
They also concluded that more work needs to be done to make cooperation agreements between the Commission's crisis management structures fully operational.
In their report, the auditors make recommendations to the member states and the Commission, including speeding up the implementation of the 2013 decision, notably by developing a strategic roadmap for the health security committee.
The auditors also want an upgrade to the 'early warning response system' and called on the EU to develop a "more structured coordination" between different Commission services for health security activities.