Yes, pretty much most PICs that support PICBasics I2CREAD & I2CWRITE will work with the 24LC1025 - that includes the 16F819.
Just for your information, EIGHT of the 24LC256's (or four 24LC1025's) will use the same number of PIC pins as just one... Two pins, SDA and SCL. The addressing of which chip you are using is done with the Control Byte you squirt down the bus and how you assign the A0 and A1 (and A2) pins on each EEPROM.
You MUST download the DATASHEET for the 24LC1025, and there on Page 7 (5.0 Device Addressing) it tells you EXACTLY the byte sequence you need to send to address the chip.
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