No, programming a bare ATtiny85 IC needs either another Arduino or an AVR programmer
Arduino as ISP $4.50
I have used an old Arduino Duemilanove using the Arduino as ISP sketch as a programmer for an ATtiny85 but any Arduino with inbuilt USB and enough GPIO could be used. There are some low cost ATmega328P or ATmega32u4 based Arduino clones that would be suitable. A Nano v3 clone costs $4.50.
Programmer $7
Something like the Pololu AVR Programmer v2.1 can be bought for $7 and used with the free Arduino development software (IDE). I have used this to program an ATtiny85. This is what I would suggest.
Development Board with USB $3
As others have commented, you can buy a "digispark" (just Google and click "shopping") for three or four dollars. This is essentially a breakout board for an ATtiny85 with a USB connector and a bootloader pre-loaded that has virtual-USB capabilities. You can program it with the Arduino IDE after using the IDE's board-manager to install Digistump's "core" definitions for the Digispark. It doesn't need another Arduino nor an AVR Programmer. I bought three on Amazon for $9 a month ago.