VCA level certifications do not have a training requirement, but VCP level certifications do (with 1 exception, mentioned below)
Having a VCA doesn't count for anything in terms of VCP level certifications.
The only way you can get your first VCP level certification with no training requirement, is the VCP6-NV for holders of certain Cisco certifications - all other VCPs have a training requirement, as does VCP6-NV if you're not Cisco certified.
-------------------------------------------------------------------------------------------------------------------------------------------------------------
Although I am a VMware employee I contribute to VMware Communities voluntarily (ie. not in any official capacity)
VMware Training & Certification blog