I'm sure your pastor would be happy to visit there, if the nursing home's management approves. God has compassion on those who are lonely and in need -- and so should we. The Bible commands us "to look after orphans and widows in their distress" (James 1:27).
I also hope you'll talk with your pastor about other ways your church could minister to those in nursing homes or other facilities in your area (such as detention centers and rehabilitation units). Many nursing homes, for example, are glad to schedule regular chapel services or church choir concerts for their residents. It's not necessarily easy to do things like this, but it can be very rewarding -- both for receipts and those who give of their time and talents.
In addition, ask God to help you be a friend to those you see when you visit your aunt. Whenever I visited my older sister in her nursing home, the staff always asked me to greet those who needed encouragement, which I was glad to do. Jesus said, "I tell you the truth, whatever you did for one of the least of these brothers of mine, you did for me" (Matthew 25:40).
All around us are people who are lonely or discouraged, not just in nursing homes, but everywhere. Ask God to help you see others through His eyes, and to point them to Jesus Christ and His unfailing love. He truly is "a friend who sticks closer than a brother" (Proverbs 18:24).