Is ADS-B intended to replace other traffic management systems?
For a short introduction on what ADS-B is and why it is called dependent see this answer.

ADS-B is a surveillance technology whereby the aircraft broadcasts information about its identity, position, altitude and velocity of the aircraft to any interesting party. The position and velocity information is derived from GNSS (GPS). 

TCAS

ADS-B is not intended to replace TCAS, although in the future it will augment TCAS. The TCAS algorithm currently only uses distance and altitude to calculate whether there is a conflict and to determine the best conflict resolution strategy. With the more accurate ADS-B position available to the system as well, the number of TCAS interrogation can be reduced (the surveillance radio frequencies are getting congested in some areas) and the performance of TCAS can be improved. A new standard for this hybrid approach was published in 2013. 

It may also be possible to have a passive TCAS-like system that does not require active interrogation but is purely depending on ADS-B. Currently a technical standard for a new Airborne Collision Avoidance System (ACAS, the generic name for TCAS) is being developed in a joint RTCA / EUROCAE committee (RTCA SC-147 / EUROCAE WG-75), which will take advantage of more data offered by ADS-B. This new standard will eventually replace TCAS II.

A 'TCAS light' for small aviation has also been developed based on ADS-B. It is called  Traffic Situation Awareness with Alerts (TSAA), a standard that was published last year. Within the next couple of years we will see manufacturers start offering products based on this standard. 

ATC

The purpose of ADS-B is neither to replace ground based ATC. But it will change the way ATC is done. ADS-B IN will improve the situation awareness of pilots; they will have a display of accurate positions of other aircraft. New procedures will allow pilots to maintain visual separation to other aircraft in marginal Visual Meteorological Conditions (VMC), where they currently would often lose sight of other traffic. A more advanced usage of ADS-B will be flight deck based interval management (FIM) where ATC will be able to instruct aircraft to 'follow that plane XX seconds behind for landing on runway YY'. A first technical standard for FIM is being worked on, it is intended to be published before the end of this year. 

ADS-B is neither a replacement for all radars, although it will allow the number of radars to be reduced. For remote areas that currently do not have radar coverage because of the associated high costs, ADS-B will be a cost effective alternative. Over the oceans, satellite ADS-B receivers will be able to provide surveillance, changing the way ATC is performed over the high seas. 

GNSS

Given the examples above, it is easy to understand the requirements on the reliability of ADS-B are substantial. Not only must the system be able to deliver accurate position data, it must be able to deliver that data with high integrity. That means, the probability that the system transmits inaccurate position data, even when there is a failure in the system, must be extremely small. To achieve this, a positioning systems must be able to do internal cross checks to detect failures and not depend on possible erroneous inputs from humans. GPS can meet the position source requirements using Receiver Autonomous Integrity Monitoring (RAIM).

The ADS-B technical specifications have all been written in such a way that they do not explicitly require GNSS to be the position source, but at the moment it is the only system that meets the requirement effectively. Most certification documents are written with the assumption that a GNSS will be used to provide position data. Theoretically one could put another system forward, but the costs of certifying that would be very high.