No, it doesn't have full TAPI support.
The PC applications software for the Cisco IOS Telephony Service—Cisco IOS TSP, a TAPI service provider software package—works as an interface between the TAPI running on Microsoft Windows and the Cisco IOS Telephony Service router. This software:
Communicates with the TAPI using the Telephony Service Provider Interface (TSPI).
Implements a required set of APIs and works with TAPI.
Enables other TAPI-based applications to provide call control to the Cisco IP phones connected to the Cisco IOS Telephony Service router.
The Cisco IOS TSP software increases your personal productivity, because you can handle phone call management from your PC without picking up the handset or dialing the numbers on the dial pad. The following features are available:
Answering incoming calls
Forwarding incoming calls to voice mail
Address book dialing (placing outbound calls from an application address and phone and address book)
Screen pop ups with caller-ID (caller ID-based screen pop ups for incoming calls; simple outgoing call placement using one-click address-book-style speed-dialing from the PC application)
Placing calls on hold
So basically the best you can do is just control you hardphone from a PC. Cisco IOS TSP software does not support media or voice. Media or voice is sent to the phone.
http://www.cisco.com/en/US/partner/products/sw/iosswrel/ps1839/products_feature_guide_chapter09186a0080153e7d.html#2175400
