Does the Play Store app verify the APKs?
Android itself checks the certificate when you update already installed APKs. It's therefore not possible to install a tampered/modified app that would need to be signed with a different certificate. It would throw a certificate mismatch exception.
Using Google Play to install a new APK for the first time is different somehow: It's signed with the developers key, but Google or a man in the middle could tamper the connection, modify it and then send it to you with a new signature. The package installer and android itself doesn't check back with google if the signature matches.
But Google Play implements the security recommendations that google gives for developers: It uses SSL and should be therefore secure. Unfortunately a wrong implementation can lead to flawed security as it happened with many apps that were tested for SSL vulnerability in this study:

http://www2.dcsec.uni-hannover.de/files/android/p50-fahl.pdf

(Not including Google Play)

Still the signature check itself is still vunrenable in 4.4:
http://www.zdnet.com/kitkat-gets-fix-for-android-app-tampering-bug-but-earlier-versions-still-vulnerable-7000022930/