Bitcoin core stopped sync'ing (without downloading all the data). How can I make it resume the synchronization?
There are pretty much two stages. The first stage is to download (catch up) the whole blockchain that has 64.4 GB.

Once you have that the verification stage checks all payments ever made.

Both these things are pretty painful but once both are done you can backup your blockchain for example to external disk and you won't ever have to do it again.

Don't delete your wallet file.  Backup your wallet.dat file.  

So delete everything, restart bitcoin, let it run for a few days and hope it doesn't get stuck.

Good luck.

Edit: do you have enough free disk space?

Edit2: don't delete the wallet